9.2 Looking for Relationships

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: 25 zero km equals 250 multiplication zero .6214 miles equals 155 miles (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:

distance in kilometers equation left hand side prefix multiplication of 0.6214 equals right hand side distance in miles

When using formulas though, we like to state first what we are calculating so the correct way to show this formula is:

distance in miles equals distance in kilometers prefix multiplication of 0.6214

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:

distance in miles equals 500 multiplication zero .6214 equals 310 .7

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.

Activity Symbol Activity: How Many Miles?

Using the word formula we have just worked out, calculate how far 350 km is in miles.

Answer Symbol

Answer

Substituting 350 for “distance in km” gives:

distance in miles equals 350 x zero .6214 equals 217 .49

So, 350 km is approximately 217 miles.

9.1.1 Pascal’s Triangle

9.2.1 Finding a Word Formula