Fractions, Decimals and Percentages
Primary 5 · Mathematics
Fractions, Decimals and Percentages
Lesson 1 of 3
12 min
Let's begin
At break time Uwase and Mugisha each buy a chapati. The two chapatis are exactly the same size. Uwase cuts hers into 2 equal pieces and eats 1 piece. Mugisha cuts his into 4 equal pieces and eats 2 pieces.
Mugisha laughs and says he ate more, because 2 is bigger than 1. Is he right? Picture it: half a chapati, and two quarters of a chapati. They are exactly the same amount of food. Mugisha only cut his into smaller pieces.
Today you will learn how one amount can wear many different fraction clothes, and how to tell quickly which of two fractions is bigger.
In the fraction 34, the bottom number 4 tells you that the whole thing was cut into 4 equal parts. The bottom number is called the denominator. The top number 3 tells you how many of those parts you are talking about. The top number is called the numerator.
Notice something important: the bigger the denominator, the smaller each piece becomes. The same chapati shared among more people means less for each person.
Take 12 of a chapati. Now cut every piece into two. What was 1 piece out of 2 is now 2 pieces out of 4. Nothing was eaten and nothing was added, so the amount of food did not change.
12 = 24
Cut everything in two again and you have 4 pieces out of 8.
12 = 24 = 48
Fractions that name the same amount are called equivalent fractions. Look at what happened to the numbers each time: the top was multiplied by 2 and the bottom was multiplied by 2. That is the whole secret.
Worked example
Write three fractions that are equivalent to 23.
Multiply the top and the bottom by 2: 2 × 2 = 4 and 3 × 2 = 6. So 23 = 46.
Multiply the top and the bottom by 3: 2 × 3 = 6 and 3 × 3 = 9. So 23 = 69.
Multiply the top and the bottom by 5: 2 × 5 = 10 and 3 × 5 = 15. So 23 = 1015.
So 23, 46, 69 and 1015 are all the same amount, just cut into different numbers of pieces.
The rule: if you multiply the numerator and the denominator by the same number, the value of the fraction does not change.
Sometimes a question looks like this: 35 is the same as how many twentieths?
Ask yourself what happened to the bottom. 5 × 4 = 20, so the bottom was multiplied by 4. Now do exactly the same to the top: 3 × 4 = 12.
So 35 = 1220.
Going the other way, you divide the top and the bottom by the same number. A fraction is in its simplest form when the only number that divides both the top and the bottom is 1.
Worked example
Simplify 2436.
Step 1 look for a number that divides both. Both 24 and 36 divide by 2. 24 ÷ 2 = 12 and 36 ÷ 2 = 18, so 2436 = 1218.
Step 2 keep going. Both 12 and 18 divide by 2 again. 12 ÷ 2 = 6 and 18 ÷ 2 = 9, so now you have 69.
Step 3 both 6 and 9 divide by 3. 6 ÷ 3 = 2 and 9 ÷ 3 = 3, which gives 23.
Step 4 is there any number apart from 1 that divides both 2 and 3? No. So you must stop. 2436 = 23, and 23 is the simplest form.
There is a faster way. Divide by the highest common factor straight away. The biggest number that divides both 24 and 36 is 12. Then 24 ÷ 12 = 2 and 36 ÷ 12 = 3, and you reach 23 in one single step.
When the pieces are the same size, the fraction with more pieces is the bigger one.
58 is bigger than 38, because 5 pieces beat 3 pieces when every piece is an eighth. We write 58 > 38.
You cannot compare pieces of different sizes just by looking at the top numbers. First you must make the pieces the same size, which means giving the two fractions the same denominator.
Worked example
Which is bigger, 23 or 34?
Step 1 find a number that both 3 and 4 go into. The multiples of 3 are 3, 6, 9, 12, 15. The multiples of 4 are 4, 8, 12, 16. The smallest number in both lists is 12. This is the lowest common multiple (LCM) of 3 and 4.
Step 2 change both fractions into twelfths. Since 3 × 4 = 12, multiply the top and bottom of 23 by 4: 23 = 812. Since 4 × 3 = 12, multiply the top and bottom of 34 by 3: 34 = 912.
Step 3 now every piece is a twelfth, so simply compare the top numbers. 9 is bigger than 8.
Answer: 34 is bigger than 23.
Worked example
Put 12, 23, 512 and 34 in order, smallest first.
Step 1 choose a denominator that 2, 3, 12 and 4 all divide into. 12 works for every one of them.
Step 2 dress every fraction in twelfths.
12 = 612 (top and bottom times 6)
23 = 812 (top and bottom times 4)
512 is already in twelfths
34 = 912 (top and bottom times 3)
Step 3 line up the top numbers from smallest to biggest: 5, 6, 8, 9.
Answer: 512, 12, 23, 34.
Tip
Two things to carry in your head:
1. Whatever you do to the top, do the same to the bottom. Multiply both by 3, or divide both by 3, and the fraction still means the same amount.
2. A big denominator means small pieces. 110 of a chapati is far less than 13 of it, even though 10 looks bigger than 3. More people sharing means less for each.
Work each one out on paper first, then tap Show answer to check.
Fill the gap: 27 is the same as how many twenty-eighths?
Answer: 28 ÷ 7 = 4, so the bottom was multiplied by 4. Do the same to the top: 2 × 4 = 8. So 27 = 828.
Write 1824 in its simplest form.
Answer: the highest number that divides both 18 and 24 is 6. 18 ÷ 6 = 3 and 24 ÷ 6 = 4. So 1824 = 34.
Which is bigger, 35 or 58?
Answer: the LCM of 5 and 8 is 40. 35 = 2440 (times 8) and 58 = 2540 (times 5). Since 25 is bigger than 24, 58 is bigger.
Keza says 46 and 23 are the same amount. Is she right?
Answer: yes. Divide the top and the bottom of 46 by 2: 4 ÷ 2 = 2 and 6 ÷ 2 = 3. So 46 = 23. Keza is right.
Put 14, 25 and 310 in order, smallest first.
Answer: the LCM of 4, 5 and 10 is 20. 14 = 520, 25 = 820 and 310 = 620. The tops in order are 5, 6, 8. So the order is 14, 310, 25.
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