Patterns occur everywhere in art, nature, science, and especially mathematics. Being able to recognize, describe, and use these patterns is an important skill that helps you to tackle a variety of different problems. This unit explores some of these patterns, ranging from ancient number patterns to the latest mathematical research. It also looks at useful practical applications.
This unit should take around eight hours to complete. In this unit you will learn about:
In Section 9.1 you will look at two examples of patterns and think about how you can continue them and describe them. Continuing with patterns in Section 9.2, we will see how when a pattern exists between different quantities we can use this to continue the pattern with word formulas.
In Section 9.3 you will use formulas to perform calculations and in Section 9.4 apply this to some situations using spreadsheets.
Once you have started to feel more comfortable with using formulas, in Section 9.5 we turn back to writing your own and techniques to rearrange formula. Section 9.6 looks at two important relationships between numbers that occur in everyday life (direct and inverse proportions), and in Section 9.7 we turn to inequalities (greater than or less than, for example).
Finally, in Sections 9.8 and 9.9 we will explore an unsolved math problem—between 2000 and 2002, a $1 million prize was offered for its solution!—and patterns seen in nature.
This unit will also help you to consolidate some of the problem solving strategies you learned in Units 6 and 7, and it introduces more notation to help you express ideas clearly and concisely.
If you would like to check your understanding of patterns and formulas before you start, give the Unit 9 pre quiz a try, then use the feedback to help you plan your study.
The quiz does not check all the topics in the unit, but it should give you some idea of the areas you may need to spend most time on. Remember, it doesn’t matter if you get some or even all of the questions wrong – it just indicates how much time you may need for this unit!
Suppose you are tiling a bathroom or kitchen, and the last row of square tiles is to be a decorative border made up of blank tiles and patterned tiles as shown below.
A friend has offered to help. How would you describe the pattern and how to arrange the tiles?
There are lots of ways of tackling this. For example, you might say that you will need some blank tiles and some patterned tiles with the “bridges” on. Start with a bridge tile, and then put a blank tile next to it. Take another bridge tile, but turn it around so that the bridge is upside down, like a smile, and put it next to the blank tile in the same line. Then put another blank tile next to the smile tile. This is the pattern: Bridge, blank, smile, blank, bridge, blank, smile …
Or you may have decided to draw a picture of the tiles or demonstrate the pattern with the tiles themselves. Whatever you do, it’s probably easier to remember and apply if you have recognized that the pattern is a four-tile repeat with the two types of tile.
The decorative border is an example of a type of geometric pattern that has many applications in art, crafts, and design.
The next example is a number pattern which appeared in China and Persia over 700 years ago, but is still used by students in mathematical and statistical problems today, and it even appears in chemistry. It is known as Pascal’s triangle and the first part of it is shown below. [ Blaise Pascal was a seventeenth-century French mathematician who studied the properties of this triangle. ]
You can continue the triangle indefinitely by following the pattern.
Each row of numbers starts and ends with the number 1. Look at each pair of numbers in the last line of the triangle. For each pair, add the two numbers together and write their sum on the line below as shown. This process generates the next row of the triangle.
You can watch a larger version of Pascal's triangle being built in this video:
The calculator can be accessed on the left-hand side bar under Toolkit.
Using the hexagonal paper provided in the attached PDF file, write down the numbers in the next two rows of Pascal's triangle. [ Save this!. You will need your triangle later in the unit. ]
Download the hexagon paper with this link:
Can you spot any patterns in the numbers in the triangle?
What do you notice about the sum of the numbers in each row?
If this pattern continues, what do you think the total for the tenth row will be?
The next two rows are 1, 6, 15, 20, 15, 6, 1, and 1, 7, 21, 35, 35, 21, 7, 1.
There are many different patterns. The 1s down the sides of the triangle are probably the easiest to spot. You can also see the counting numbers 1, 2, 3, 4, 5 in the diagonal rows next to the sides. In the next diagonal row, there is the sequence 1, 3, 6, 10, … . These numbers are known as the triangular numbers because they create triangular patterns as shown here. [Do not worry if you did not spot all these patterns.]
Pascal’s triangle is symmetrical, too. If you draw a vertical line down through 1, 2, 6, … , one side of the triangle is a mirror image of the other. [ The pattern in the powers of 2 shown in the next diagram suggests that the first total, 1, might be written as . Check on your calculator to see if this is correct. Does any number to the power 0 give you 1? ]
As you look at the diagonal row of triangular numbers, if you add any two adjacent numbers together, you get the square numbers. For example, , which is the same as , and , which is the same as , and so on.
Now, if you add the numbers in each horizontal row, you get the following pattern: 1, 2, 4, 8, 16, 32, 64, … . The total for the next row is double the total for the current row. Assuming this pattern continues, the total for row 8 will be 128, row 9 will be 256, and row 10 will be 512. You might have noticed that these numbers can also be written as … .
What do these two examples have to do with mathematics? Well, recognizing patterns in shapes, sets of numbers, processes, or problems and noticing what is the same and what is different about situations often makes a task easier to solve. You saw how recognizing the tiling pattern makes it easier to remember, and by using the number patterns in Pascal’s triangle, you could work out the sum of each row without adding the individual numbers.
If you can spot a pattern and then describe what happens in general, this can lead to a rule or formula. If you can prove that this rule will always work, it can be used elsewhere. For example, if you can work out the general process for calculating a quarterly electricity bill and then give these instructions to a computer, many electricity bills can be generated, printed, and sent out in just a few minutes.
In this section, we are going to consider the relationship between quantities in two practical situations and see how to describe these relationships by writing down a general rule or a word formula.
Suppose you are planning a visit abroad. Your map marks all the distances between places in kilometers rather than miles. How can you work out what these distances are in miles?
The first type of information you will need is how long one kilometer is, measured in miles.
You may use the Internet to verify that 1 km is equivalent to approximately 0.6214 miles.
The next step is to work out the mathematical process to change kilometers into miles. You may be able to see how to convert from kilometers into miles immediately, but, if not, try to visualize a few simple examples. [ Drawing a diagram often helps! ]
Each km is the same as 0.6214 miles as shown below.
So, if the distance is 3 km, you will have 0.6214 miles three times; if the distance is 10 km, you will have 0.6214 miles 10 times; if the distance is 250 km, you will have 0.6214 miles 250 times, and so on.
You can write this last example down mathematically as: (to the nearest whole number).
Notice that in each of the examples above, the process for calculating the number of miles was the same: multiply the number of kilometers by the conversion rate of 0.6214. This technique will work for any distance and can be written down concisely as the following word formula:
When using formulas though, we like to state first what we are calculating so the correct way to show this formula is:
You can use this formula to convert any distance in kilometers into miles. For example, suppose you wanted to convert 500 km. Using the formula and replacing “distance in kilometers” by 500 gives:
So, 500 km is equivalent to approximately 311 miles.
In this example, you used the formula by replacing the phrase “distance in kilometers” on the right side of the equation by the corresponding value, 500. This is known mathematically as substituting the value into the formula.
Using the word formula we have just worked out, calculate how far 350 km is in miles.
Substituting 350 for “distance in km” gives:
So, 350 km is approximately 217 miles.
Suppose you are visiting Europe, and you want to exchange some money from dollars into euros. In 2011, one agency offered an exchange rate of $1.00 to €0.74 and did not make any additional charges. [ You can look up current exchange rates and fees on the Internet, or from some newspapers, banks, or travel agencies. ]
(a) How many euros would you get for $5? How many euros would you get for $10?
(b) Write down a word formula that you can use to convert dollars into euros. [ What would happen to the formula if there were also a fee per transaction? ]
(b) To change dollars into euros, multiply the number of dollars by the exchange rate of €0.74. The word formula to represent this is:
If there were also a fee per transaction this would have to be added to the formula.
(c) Check that your formula works by using it to convert $5 into euros.
(c) Substituting 5 for “number of dollars” gives the number of euros: . This agrees with the answer in part (a).
These two examples illustrate how a word formula can be used to summarize a mathematical process such as converting units of length or currencies. Once a formula has been derived, it can then be used in other situations, both for calculations by hand or by computer—for example, for currency transactions in a bank. You will be able to practice writing your own formulas in Sections 9.4 and 9.5.
Now let's look at using formulas.
In the last section, we considered how a formula could be built and then how it could be used. This section considers some more complicated formulas that have already been developed and are used in a variety of different situations—cooking, health care, business, and archaeology. We hope that these examples illustrate some of the broad applications of math and how mathematical relationships can be used in making decisions. As you work through these examples, you might consider where else math could be used in each of these topics.
When you are trying to solve a real-life problem mathematically, you often use formulas that have already been developed, so it is important to have ideas on how to apply them.
This section will help you do that, and you might want to make your own list of tips in your math notebook as you work through this section and compare it with the box at the end of this section.
Try asking your friends and family if they have come across any formulas and what they are for.
The calculator can be accessed on the left-hand side bar under Toolkit.
The time to cook a fresh chicken depends on its mass, commonly called weight in everyday language, as given by the following formula:
Roughly how long will a chicken with a mass of 2.2 kg take to cook?
To use the formula, you need to substitute the mass of the chicken into the right side of the equation and then work out the resulting calculation. However, the formula asks for the mass in grams, so the first step is to convert 2.2 kilograms (kg) into grams (g).
Since there are 1000 g in 1 kg, .
Substituting 2200 for the mass in grams gives: .
Carrying out the multiplication and division first gives (remember PEMDAS!): .
So, the cooking time would be about 125 minutes, or 2 hours and 5 minutes. If you prefer, you can use a calculator, but remember to make an estimate first so that you may check that your answer is reasonable.
How long would it take to cook a chicken with a mass of 1.6 kg?
First, we must convert 1.6 kg into grams: .
The cooking time can then be calculated in minutes: minutes.
The chicken would take about 95 minutes, or 1 hour and 35 minutes.
The calculator can be accessed on the left-hand side bar under Toolkit.
The body mass index (BMI) is sometimes used to help determine whether an adult is under- or overweight. It is calculated as follows:
Although care needs to be taken in interpreting the results (for example, the formula isn’t appropriate for children, old people, or those with a very muscular physique), a BMI of less than 20 suggests the person is underweight, and a BMI of over 25 suggests the person is overweight.
In this formula, the units have been included in the expression on the right side of the equals sign. It is important to change any measurements into these units before you substitute the values into the formula. (No units have been included for the BMI on the left side, in line with current health care practice.)
If an adult man is 5 feet, 10 inches tall and weighs 185 pounds, calculate his BMI and decide whether he is overweight.
The formula needs the weight in pounds and the height in inches. First, convert the height into inches. Since there are 12 inches in a foot: .
Substituting the weight and height into the formula for the BMI gives:
Since his body mass index is over 25, the man is probably overweight. [ What would be a healthy weight for him? Use a BMI of 25 and his height and see if you can work backwards - you should come out with about 174 pounds. ]
One of the advantages of identifying the general features of a calculation and then describing it mathematically is that the formula can then be used in either a computer or a calculator program to work out different calculations quickly and efficiently. Many utility suppliers (gas, water, electricity, telephone) have rates based on a fixed daily (or monthly or quarterly) charge and a further charge based on how much you have used during the billing period.
For example, in 2005, a mobile phone network charged $25 per month for 30 minutes (or less) of phone calls. Extra calls above the 30 minutes were charged at 20¢ per minute.
Assuming that more than 30 minutes were used per month, the formula for the total monthly cost can be expressed by the following word formula.
Suppose that in one month, 75 minutes of phone calls were made. Explain how you would calculate the cost for the extra minutes (above the 30-minute allowance) and then how to calculate the total cost for that month. Can you explain how the formula has been derived?
As 30 minutes are included in the $25 charge, the number of minutes that are charged separately is .
Each extra minute costs 20¢, so 45 extra minutes will cost .
So, the total charge for that month is .
The formula says that the fixed charge is $25, and then each minute in excess of the 30 minutes allowed costs 20¢.
On this phone bill, what would the monthly charge have been if the total time for the calls were the following?
(a) 48 minutes
Substituting 48 for the total number of minutes gives the total cost in dollars as:
So, the bill for the month is $28.60.
Note how this calculation is set out with a concluding sentence that answers the question precisely.
(b) 25 minutes
The formula only applies if more than 30 minutes of calls are made, so it cannot be used in this case. This is an important step in using formulas—check that they apply in your situation before using them. Here, the charge is $25 for up to 30 minutes of calls, so the charge for this month is $25 overall.
If you have a similar deal you could work out a formula that fits yours and use it to check your bills!
The calculator can be accessed on the left-hand side bar under Toolkit.
In parts of the world, footprints from prehistoric human civilizations have been found preserved in either sand or volcanic ash. From these tracks, it is possible to measure the foot length and the length of the stride. These measurements can be used to estimate both the height of the person who made the footprint and also whether the person was walking or running. This can be done by using three formulas.
Note that no units have been included in these formulas, so it is important to make sure that the same units, for example centimeters, are used throughout the calculation. If the value of the relative stride length is less than 2, the person was probably walking, and if the value is greater than 2.9, the person was probably running.
From one set of footprints, the length of the foot is measured as 21.8 cm and the stride length as 104.6 cm. What does the data suggest about the height and the motion of the person who made these footprints?
Break the problem down into simpler parts. Start by asking yourself what you need to know to tell if the person was running or walking.
So, the estimated height of the person = = 153 cm (to the nearest cm). [ How do you think these formulas have been derived? How reliable do you think these estimates are? ]
To work out if the person was running or walking we need to know the associated relative stride length. This is given by:
So now we need to calculate the hip height and then substitute it into the relative stride length formula.
The hip height = = 87.2 cm.
So, the relative stride length (rounded to 1 decimal place).
As the relative stride length is less than 2, the person was probably walking.
Notice that the relative stride length has no units. The stride length and the hip height are both measured in cm, and when we divide one by the other, these units cancel each other out, just like canceling numbers. If you go on to study physics or other sciences, you will find many examples like this.
Notice that to find the relative stride length here, we tackled the problem by using the two formulas step by step, but you could also have combined these two formulas into one:
This new formula could be used directly.
A student used the formula: to calculate the relative stride length in the footprints activity.
Use this formula to work out the relative stride length and check that your answer agrees with the result in the last activity. The student entered the key sequence for the calculation as . This gave an answer of approximately 570, so the student concluded that the person was probably running very fast. Can you explain where the student made the mistake?
The calculator will perform this calculation from left to right, using the PEMDAS rules (order of operations), which treat multiplication and division as equally important. So, it will first divide 104.6 by 4 to get 26.15, and then multiply by 21.8 to get approximately 570. However, this is not the correct calculation from the formula.
[ Can you explain why these two calculations are equivalent? ] The stride length (104.6) should be divided by , so the calculation should be or alternatively . Enter both these expressions into your calculator and check that these both give the correct answer.
Note that the student should have been expecting an answer between about 0 and 5, so an answer of 570 is suspicious! Also if the student had worked out an estimate for the answer first, the mistake might have been noticed.
Now, look over the formulas you have used in this unit and also any others that you might be familiar with from your home, work, or hobbies. What are the common steps in using a formula? If you were asked to write down a list of tips for using a formula, what would you say? Here are a few suggestions—feel free to add your own!
Although there may be many occasions when you are given a formula, sometimes you may need to devise your own, for example if you use a spreadsheet on a computer at home or at work. This section looks in more detail at the process of devising a formula.
Part of a spreadsheet that has been constructed to record monthly income and expenditures is shown below. It is similar to a balance sheet that you might draw up by hand and includes the monthly income and expenses, the totals, and the overall balance. It is used to keep track of expenditures, particularly to try and ensure that the balance remains positive. However, the spreadsheet has been stored on a computer rather than on paper and is updated regularly. One of the advantages of using a computer spreadsheet is that you can insert formulas into the spreadsheet to carry out calculations automatically, without using a calculator.
A spreadsheet is made up of rows and columns of cells. The columns are identified by letters and the rows by numbers. This enables you to identify each cell in the spreadsheet. For example, the number 221.12 is in column B, row 11, so this cell is B11.
Cells can be found from their reference by looking down the column and across the row. So, cell A3 can be found by looking down column A and across row 3. This cell contains the word “Salary.” Notice that cells can contain either text or numbers.
Use the spreadsheet above to answer the following questions.
(a) What is contained in the following cells?
(i) The cell that is in both column A and row 5 contains the words “Total Income.”
(ii) Down column A and across row 12, cell A12 contains the word “Other.”
(iii) The cell that is in column B, row 15 contains the number 166.72.
(iv) The cell that is in column B, row 1 contains the heading “Amount ($).” This indicates that the entries in this column are amounts of money and that they are measured in $. This heading could also have been written as “Amount in $.”
(b) What is the reference for the cells that contain the following?
(i) The number 1585.18
(ii) The word "Food"
(i) 1585.18 is in column B and row 13, so its reference is B13.
(ii) “Food” is in column A and row 9, so its reference is A9.
Now, look again at the spreadsheet and see if you can work out what information it shows. For example, if you look at row 3, this shows that the monthly salary is $1,700.56. Although cell B3 only contains the number 1700.56, you know that this is measured in dollars from the heading in cell B1. Overall, the spreadsheet shows the items that make up the monthly income and where money has been spent over the month. If you were keeping these records by hand, you would then need to calculate the total income, the total expenses, and the balance.
For example, to find the total income for the month, you would need to add the salary of $1,700.56 to the other income of $51.34. This gives the total income of $1,751.90. In other words, to calculate the value in cell B5, you need to add the values in cells B3 and B4. This can be written as the following formula:
Formulas have also been used to calculate the total monthly expenses and the balance. If you were working these calculations out by hand, what would you do? Check by comparing your answers with the values in cell B13 and B15.
Try to write down the formulas for these cells in the form “value in B13 = …”
To calculate the total monthly expenses, you need to add the individual expenses on “Rent,” “Food,” “Transportation,” “Regular Bills,” and “Other.” The formula will be:
To find the balance, you need to take the expenses away from the total income. So, the new formula will be:
Did your formulas give you the correct answers?
To put these formulas in the spreadsheet, you can type the formula directly into the relevant cell, starting with an equals sign to show that you are entering a formula rather than a word.
Notice that in cell B13, a shorthand form for the sum has been entered. You could have typed in “= B8 + B9 + B10 + B11 + B12” but it is also acceptable to use the shorthand form, “= SUM(B8:B12)”, which is shown here. This instruction tells the computer to add the values in all the cells from B8 to B12.
One of the advantages of using a spreadsheet is that if you change some of the numbers, all the calculations that use that particular number will be automatically updated to reflect the change. For example, in this budget, the amounts for the salary, rent, and the regular bills are likely to remain the same from one month to the next and may only be updated once or twice a year. However, the amounts for food, transportation, other bills, and other income probably will change from month to month. These values can be changed easily on the spreadsheet and the revised balance produced immediately.
The formulas used in a spreadsheet can be displayed as shown in the diagram below. Note that * is the notation for multiplication and that all entries have been rounded to the closest cent. This example shows Maryland state sales tax, which was 6% in 2011 (6% as a decimal is 0.06).
(a) The values in columns C and D will be displayed to two decimal places as they represent an amount of money. What values will be displayed in cell C3 and cell D3?
(a) The formula in C3 is: .
So, (rounded to 2 decimal places).
The value in D3 is obtained by adding together the values in B3 and C3.
So, = 17.48.
(b) What do you think is being calculated in the cells in column D? Can you suggest a suitable heading for this column, to be entered into D1?
(b) Column D represents the total cost of the item plus sales tax. A suitable heading might be “Total.” You can think of other correct titles such as “Final Price ($).”
(c) In cell D5 the sum of the values in D2, D3, and D4 will be calculated. Write down a formula that could be entered in cell D5. What does this value represent?
(c) The sum can be found by adding together the values in cells D2, D3, and D4. The formula “=D2+D3+D4” could be entered into cell D5. (Note that you do not have to use the formulas that are present in each of the cells you are adding. Just the cell title is sufficient.) Alternatively, you could use “SUM(D2:D4)”, which sums up the cells from D2 to D4 as well. The resulting entry represents the total cost (including Maryland sales tax) of the radio, kettle, and fan together.
With these changes, and the title “Total ($)” typed into cell C5, the spreadsheet will look like the following.
In the last section, you worked out formulas that could be used in a spreadsheet. This section gives you more practice in deriving formulas both by looking at number tricks and rearranging existing formulas.
Try this number puzzle: “Think of a number. Add 5. Double it. Subtract 8. Divide by 2. Take away the number you first thought of. Add 4.” [ If you use your calculator, remember to press the Equals sign after each instruction. ]
Now if 1 represents the letter “A,” 2 represents the letter “B,” 3 represents the letter “C,” and so on, work out the letter represented by your answer and write down the name of an animal beginning with this letter.
Start by choosing 3 as the initial number.
The instructions work out as follows.
|Starting with 3|
|Divide by 2:|
|Subtract your number:|
You will find that you always get 5, whatever number you start with. This gives the letter E. Did you choose an elephant?
To see how this trick works, read through the instructions from before. Because you could have thought of any number, replace this unknown number by a thought bubble like this:
Keep the thought bubble as part of the new number that you get at each stage.
This shows that the numerical answer is always 5—it does not depend on which number was chosen first. So the letter of the alphabet chosen is E. There are not many animals with names beginning with E and most people do think of an elephant first, but you might be unlucky if someone chooses an emu or an eel!
Rather than using a cloud to represent the number and explaining the trick pictorially, you could write down expressions for the answer at each stage and use either a word (for example, number) or a letter (for example, n) to represent the number. Write out the trick again, expressing it with the unknown number n.
|Think of a number:|
|Divide by 2:|
|Subtract your number:|
Try the following trick several times. Think of a number between 1 and 10. This will work with numbers greater than 10, but the restriction is to keep the arithmetic manageable. [ Try making up your own number tricks. What makes a good trick? ]
Multiply by 4. Add 6. Divide by 2. Subtract 3. Divide by 2, and your answer is?
Write down the number you first thought of and your answer. What do you notice? Can you explain why this happens, either by using a diagram or by writing down the expressions for the answer at each stage?
You should find that this time the answer is always the number you chose at the start. The expressions for the answer at each stage are shown below.
|Think of a number between 1 and 10:|
|Multiply by 4:|
|Divide by 2:|
|Divide by 2:|
Now try the following.
Think of a number. Add 4. If my answer is 11, can you work out what number I was thinking of?
You might have said “What number do I have to add on to 4 to get 11?” or perhaps “If I take away 4 from 11, what number do I get?” In both cases, you should have arrived at the answer 7.
In the second method, “subtracting 4” undoes the “adding 4” in the original instructions.
This process can be illustrated by a “doing-undoing diagram.”
In the doing part of the diagram, start with the number and write down the operations applied in turn until you get the answer. Here there is just one operation, “add 4.”
For the undoing part of the diagram, start on the right with the answer, in this case 11. Then work back toward the left, undoing each operation in turn until you find the starting number. In this case, “subtract 4” undoes “add 4” and . So, the number first thought of was 7. Notice how the arrows indicate the direction to read the diagram.
The next activity gives you some practice with doing and undoing. You will find these techniques helpful for rearranging formulas later. Part (c) involves two stages.
Try to work out what number I was thinking of in the following problems. You may find it helpful to use some doing-undoing diagrams.
(a) Think of a number, subtract 3, and the answer is 2.
(b) Think of a number, multiply by 5, and the answer is 35.
When you have to deal with more than one operation, just take each step in turn. Write down the doing diagram and draw the undoing one underneath, working backward to undo each operation.
Addition and subtraction.
Multiplication and division.
Squaring and taking the positive square root.
Think of a number. Add 5. Multiply by 3. Subtract 4.
If the answer is 17, can you work out what number I was thinking of?
Here there are several steps, so to find the number, it will be necessary to undo each of these steps in turn, starting with the last step. Draw the doing diagram and then the undoing one beneath it.
You might like to go back to the BMI activity (Section 9.3.2) now and see if you can work out the weight at which the man would be classed as normal weight.
We will now look at how we can use this same technique to change formulas.
The same technique can be used to change formulas. For example, earlier in the unit, you found a formula to convert dollars into euros:
However, what if you were from Europe and wished to convert euros into dollars, say while you were shopping on vacation?
Then you would need a formula for the number of dollars based on the number of euros. We can tackle this by drawing the “doing and undoing” diagrams for this situation:
So, starting at the right of the undoing diagram, the formula for converting euros into dollars is:
To change kilometers into miles, you used the formula:
Starting with “distance in kilometers,” draw a doing diagram to get the distance in miles. Draw the undoing diagram and write down the formula for changing miles into kilometers.
Your formula should start “distance in kilometers =.”
Recall the formula for the monthly cost in $ of a mobile phone that we used earlier:
The owner wishes to stick to a monthly budget of $45. Starting with the “total number of minutes,” draw a doing diagram to show the operations to find the “monthly cost.” Put the monthly cost equal to $45 and then draw an undoing diagram to work out how many minutes of phone calls can be made if the monthly cost is $45.
With practice, this process of using a doing-undoing diagram becomes second nature. In some cases, it can be useful if you need a formula in a different form. However, for complicated formulas, a different approach (which you will need if you continue your mathematical studies) is often used.
In this section, we consider two important types of relationships that occur frequently in real life: direct and inverse proportion.
The first type of relationship is known as direct proportion. Two quantities are said to be directly proportional to each other if when one doubles, triples, and quadruples, the other also doubles, triples, and quadruples. For example, if you buy three times as many items as usual, you would expect to have to pay three times as much money (unless there were some special offer available) because the price is directly proportional to the number of items bought.
Which of the following quantities are in direct proportion?
Select the two correct answers from the following:
(a) The number of kilometers and the equivalent number of miles.
(b) The number of euros that are exchanged for dollars.
(c) The monthly cost of a cell phone and the minutes used for talking in Section 9.3.3.
The correct answers are a and b.
The correct answers are (a) and (b).
Parts (a) and (b) are both examples of direct proportion. If you multiply the number of kilometers by any factor, the number of miles will also change by this factor.
For example, if you double the number of kilometers, the number of equivalent miles will also double, and similarly for exchanging currency.
If you triple the number of dollars you exchange, you would expect to get three times as many euros. However, part (c) is not a directly proportional relationship. If you use 30 minutes, the charge is $25. But if you double the time used to 60 minutes, the charge is $31, and the price has not doubled.
You can tackle problems involving direct proportion in many ways. Using a formula is not always the easiest way, as the following example shows.
Suppose that in a coffee club at work, a group of people share a carton of milk each day, but provide their own coffee. The carton contains enough milk for 12 cups of coffee. If the carton costs 72¢, how much should someone who has four cups of coffee with milk pay?
This person had of the milk, so the person should pay of the cost of the milk, which is 24¢.
Alternatively, you can work out the cost of milk for one cup of coffee as a first step. [ Note how the problem has been simplified by considering just one cup of coffee first. ]
Since there is enough milk for 12 cups, one cup will cost 72¢ ÷ 12 = 6¢.
So, four cups will cost 4 × 6¢ or 24¢.
A recipe for a vegetable curry for four people requires 24 oz of rice. By first working out how much rice one person requires, calculate how much rice would be needed for nine people.
A patient needs to have 600 mg of a drug each day, given in three doses. Each tablet contains 100 mg of the drug. How many tablets are needed for each dose?
One tablet contains 100 mg of the drug. [ Remember to check that your answer seems reasonable. ]
Six tablets will contain 600 mg of the drug. The patient will need to take six tablets spread over three doses.
So, the number of tablets in each dose will be .
You can also use a formula here: .
Use the formula to see if you get the same answer.
The second type of relationship is known as inverse proportion.
Suppose you have decided to hire a taxi to take a group of colleagues from work to the train station. If the taxi charges a set fee for the journey, then the more people who go in the taxi, the less each person has to pay: If two people go, each pays half the cost; if three people go, each pays a third of the cost; and if four people go, each person pays a quarter of the cost. This is an example of inverse proportion. As one quantity (the number of people) doubles or triples, the other quantity (the cost) halves or is reduced to a third of the original value. In other words, if you multiply one quantity by a factor, the other quantity is divided by that factor.
The cost per person can be calculated directly or by using the following formula:
For example, if the fare is $23.00 and four people go in the taxi, each person will have to pay .
Two quantities are said to be inversely proportional if their product is a constant. For example, suppose you drive 30 miles and it takes you an hour to get there. Your average speed for the trip is 30 miles per hour. Now if you doubled your speed to 60 miles per hour, it would take you half as long (that is, half an hour) to get there; if you halved your speed, it would take twice as long, and so on. In each case, the following relationship holds:
This shows that for a given distance, speed and time are inversely proportional to each other.
This formula can also be written as: .
For driving the 30 miles above, what is the average speed if the trip took hours?
Substituting the distance as 30 and the time as the decimal 1.5 in the formula above gives:
So, the average speed for the trip is 20 miles per hour.
In this unit, there have been three occasions when checks have been made to see whether the result is greater than or less than some other value. The first case was in calculating the BMI and determining whether the person was overweight or underweight; the second case was in determining from footprints whether a person was walking or running; and the third case was in checking whether a phone had been used for more than 30 minutes. Checking whether values are greater than or less than some limit happens frequently, particularly with safety issues, but elsewhere, too. For example, medicines may have to be stored at a temperature of 77°F or less. Child train tickets can be bought for children who are 5 or more years old but less than 16 years old.
Rather than writing out “greater than” or “less than,” shorthand notation is often used as shown below.
[ If you have difficulty interpreting these symbols, you can think of them as arrows that point to the smaller number or remember that < looks like an L, which stands for less than. ]
> greater than
≥ greater than or equal to
< less than
≤ less than or equal to
The symbols can be read from left to right. For example, 11 > 9 is read as “11 is greater than 9” and vacation cost in dollars < 1,000 is read as “vacation cost is less than $1,000.”
To use the symbols in your own writing, decide what you want to say first, then use the symbol. For example, since 10 is greater than 5, this could be written as 10 > 5 or since on the number line, lies to the left of , is less than , and this could be written as < .
Similarly, the instructions for the medicine could be written as: Medicine temperature in °F ≤ 77.
Sometimes, you may find it helpful to draw a number line to visualize this kind of information. For example, the ages which children are eligible for the child train fare are from their fifth birthday up to but not including their 16th birthday. This means that the age has to be greater than or equal to 5 and less than 16. [ Note how drawing a diagram here helps you to see what is happening. ]
The empty circle means that this number (16) is not included and the filled-in circle means that this number (5) is included in the interval. This can be written as: 5 ≤ age for child train fare < 16.
Note carefully the format in which this last inequality is written. The variable that is being described, that is the age for a child train fare, is always in the center when defining a range of values with an upper and lower limit as we have here. This is the math convention that we follow with ranges of values.
For this activity, you should write your answers to the prompts on a separate sheet of paper. Once you are finished, compare your answers to the answers here. Don't worry if your wording does not match exactly.
(a) Place the correct symbol ( < or > ) in the blank spaces below.
(i) 4 __ 7
(ii) 18 __ 10
(iii) 3 __
Write the inequality using words first and then convert this to the correct symbol.
(a) Compare your answers to the correct answers here.
(i) 4 < 7
(ii) 18 > 10
(iii) 3 >
(b) Evaluate what the following mathematical statements mean. Write your answers as full sentences.
(i) Balance in account > 0.
(ii) Speed (in mph) on highway ≤ 65.
(iii) 18 ≤ age (in years) ≤ 50.
(b) Compare the answers you wrote down to the correct answers here. They may be worded differently, so make sure that they have the same meaning.
(i) The balance in the account is greater than zero, so there is credit in the account.
(ii) The speed on the highway is less than or equal to 65 miles per hour.
(iii) The age in years is between 18 and 50, inclusively.
(c) Rewrite the following sentences as statements using the inequality symbols.
(i) The refrigerator temperature should be less than 40°F.
(ii) There must be at least five people on the committee.
(iii) For an ideal weight, a person’s BMI should be greater than 20 and less than 25.
(c) Compare the answers you wrote down to the correct answers here. They may be worded differently, so make sure that they have the same meaning.
(i) Refrigerator temperature (in ºF) < 40.
(ii) Number of people on committee ≥ 5.
(iii) 20 < BMI < 25
Inequalities are also used in computer programs to check whether conditions have been fulfilled. For example, if the balance in your bank account is negative, you may be prevented from withdrawing money from a cash machine.
The more practice you can get at inequalities, the easier they will get so try out this game.
Much of the mathematics you have looked at in the course has been used to help people solve problems for centuries. However, there is much more to mathematics than that. Exciting and new developments are being made all the time, and there are many problems that mathematicians have not been able to work out. This section takes a look at one of these problems. [ It was unsolved at the time of writing! ]
In 1742, a Prussian mathematician, Christian Goldbach, made the following conjecture:
All positive even integers bigger than or equal to 4 can be expressed as the sum of two primes. [ A conjecture is a theory that has not yet been proved to be true. ]
The above statement contains a lot of mathematical terminology, so before going further, here are some helpful definitions.
An integer is a whole number, negative, zero, or positive; for example, , 23, 0, and 281 are all integers.
An even integer is one that can be divided exactly by two; for example, , 0, 42, 128, or 1002.
The last digit of an even integer is always 0, 2, 4, 6, or 8. Integers that are not even are odd. [ A lot of encryption processes depend on prime numbers. ]
A prime number is a whole number greater than 1 that can only be divided exactly by 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59.
Now, break down Goldbach’s conjecture into steps:
Try to express the following numbers as a sum of two primes.
4, 12, 28, 40, 62
For each number, try to find as many combinations of two primes as you can. Do you think Goldbach’s conjecture is true?
It can help if you look at the list of prime numbers systematically, starting with 2. Is there a number in the list that you can add to 2 to get the number you want? Then look at 3, and so on. You should be able to find the following combinations:
Are you convinced that Goldbach’s conjecture is always true?
Although the conjecture does work for the few numbers we have tried here, that does not necessarily mean that it is going to work for all the even integers greater than or equal to 4. The even integers go on forever, so it is not possible to check them all individually either by hand or computer—some other way of proving the result is needed.
Although higher branches of mathematics have been successful in tackling similar problems, this one is turning out to be particularly tough. So tough in fact that in March 2000, the publishers Faber and Faber offered a $1 million prize to anyone who managed to prove the conjecture by March 2002. No one did. So even though many great mathematicians have attempted a proof, over 250 years later, it still hasn’t been done.
This website provides a Goldbach's Conjecture calculator if you would like to try out some even numbers for yourself. It will tell you the pair of primes that add up to your number.
You might find that reassuring the next time you get stumped with a mathematics problem—being stuck is a natural state for a mathematician! It’s often the time when great discoveries are made or when the best learning takes place.
If you take a cauliflower and break off one of the florets, the floret appears to be the same shape as the original cauliflower but on a much smaller scale. You can probably see that in the picture of a related vegetable below called the Romanesco.
This idea is known as self-similarity in math and occurs throughout nature—the frond of a fern looks like the whole fern, a branch of a tree splits into further branches which look like trees themselves, and so on.
Self-similarity now forms part of an area of mathematical research known as fractal geometry. Beautiful patterns can be generated by fractal formulas, such as the Barnsley fern and the fractal below. Fractal geometry also has many practical applications in the physical sciences, medical research, economics, and computing, particularly in compressing images.
You can generate part of a fractal yourself by using Pascal’s triangle, which you constructed earlier in the unit. If you shade in (or cover) all the odd numbers in the triangle, a pattern starts to emerge as shown below.
[ Waclaw Sierpinski (1882–1969) was a Polish mathematician. ] This is part of a fractal known as Sierpinski’s triangle. It can also be generated by starting with a triangle, splitting it into four smaller triangles, and removing the middle one. Then split each of the remaining triangles into four smaller triangles and remove the middle one. This results in something like the sequence below.
Below are some exercises that will help you continue to develop your ability and check to make sure you understand the concepts discussed in this unit. Be sure to write your work out in your math notebook so that you can refer to it later if necessary.
Describe the patterns that you can see in these two lists of numbers.
(a) 1, 3, 5, 7, 9
(b) 1, 4, 9, 16, 25
(a) Starting at one, to find the next number add 2, or; a list of odd numbers starting from one.
(b) A sequence of square numbers starting with the square of 1, then the square of 2 and so on.
a) A pedometer counts the number of steps that you take over a given time period but you want to work out how far you have walked in yards. If your normal stride length is 0.75 yards write down a word formula that will allow you to calculate for any given number of steps how far you have walked.
b) Your electricity is charged at 15 cents per unit with a monthly standing charge of $2.50. Write a word formula that allows you to calculate your annual electricity bill in dollars.
The formula to convert temperatures from Fahrenheit to Celsius is:
Using this formula (remember PEDMAS) convert the following temperatures from Fahrenheit to Celsius.
a) 45 ºF
b) -3 ºF
c) 110 ºF
a) = 5.6 ºC (to 1 d.p.)
b) = -19.4 ºC (to 1 d.p.)
c) = 43.3 ºC (to 1 d.p.)
Using the technique of doing and undoing work out how to convert from Celsius back to Fahrenheit.
So the formula to convert from Celsius to Fahrenheit is: