6.1.7 True or False?

True or False?

This activity is different from the numerical activities you’ve seen so far: it asks you to consider some general statements about fractions.

Activity symbol Activity: True or False?

Read the following statements and decide whether they are true or false.

(a) “Only a fraction of the group were on time” always means that less than half the people were on time.

Hint symbol

Discussion

Can you think of a fraction that is larger than one half?

Solution symbol

a. 

True


b. 

False


The correct answer is b.

Answer

(a) False. The fraction could be three divided by four or seven divided by eight, which are both bigger than one half. Be careful! The use of the word “only” may suggest to you that it is a small fraction, perhaps less than one half, but this could be a wrong interpretation.

(b) You can write any fraction in a decimal form.

Hint symbol

Discussion

What operation does a fraction bar represent?

Solution symbol

a. 

True


b. 

False


The correct answer is a.

Answer

(b) True. A fraction can be thought of as a division problem. For example, seven divided by eight is the same as seven division eight equals 0.875. However, some decimal fractions do not stop; instead they have a repeating set of digits, such as equation left hand side two divided by seven equals right hand side 0.285714285714 times horizontal ellipsis.

These are known as repeating decimals. They are accurately represented by placing a bar over the repeating set, like this: equation left hand side two divided by seven equals right hand side 0.285714 minus.

An alternative representation for a repeating decimal is to place a dot over the first and last digit of the repeating set.

Often, these decimal numbers are rounded, so you might see two divided by seven rounded off to 0.29 or 0.286.

Keep in mind that rounded values, while useful for some purposes, are not the same as accurate representations.

(c) Fractions always have a value of less than one.

Hint symbol

Discussion

Could the numerator (top number) of a fraction be larger than its denominator?

Solution symbol

a. 

True


b. 

False


The correct answer is b.

Answer

(c) False. A positive fraction in which the numerator is greater than the denominator has a value greater than 1. For example, seven divided by three (seven-thirds) has a value greater than 1, because it means seven division three, which is greater than 2.

Brain Stretcher

(d) A number can always be written as a fraction.

Hint symbol

Discussion

Can you write a whole number as a fraction? Can you write a decimal number such as 0.375 as a fraction?

Solution symbol

a. 

True


b. 

False


The correct answer is b.

Answer

(d) False. Some numbers, but not all, can be written as fractions. Whole numbers, such as 8, can be written with a denominator of 1, like this: eight equals eight divided by one. So, if you have a finite number that stops after a certain number of decimal places, then it can be written as a fraction. For example: equation sequence 0.375 equals 375 divided by 1000 equals three divided by eight. These are not approximations or repeating fractions: 0.375 is exactly equal to three divided by eight.

Even decimals that have a repeating sequence can be written as fractions. For example: equation left hand side zero full stop times three minus equals right hand side one divided by three and equation left hand side zero full stop times 36 minus equals right hand side four divided by 11. However, it can be proven that there are some numbers, such as pi (pi) and Square root of two that cannot be written as fractions. These are known as irrational numbers. In math, the word irrational means it cannot be written as a ratio of two integers.

6.1.6 Fractions of a Group

6.1.8 Mixed Numbers