Mathematics notes, unit 2
Fractions, Decimals and Percentages
Textbook Mathematics Year 4 (DLP), pages 67–104
What this chapter covers
- You can change mixed numbers to improper fractions and improper fractions to mixed numbers (2.1.1)
- You can add mixed numbers, proper fractions and whole numbers, with answers in the simplest form (2.1.2)
- You can subtract fractions and mixed numbers, including from a whole number (2.1.3)
- You can solve sums that mix addition and subtraction of fractions, working from left to right (2.1.4)
- You can find a fraction of a quantity, including a mixed number of a quantity (2.1.5)
- You can add, subtract, multiply and divide decimals up to three decimal places, and multiply and divide by 10, 100 and 1 000 (2.2.1 to 2.2.4)
- You can change fractions to percentages and percentages to fractions, and find a percentage of objects (2.3.1, 2.3.2)
- You can solve problems with fractions, decimals and percentages in daily life (2.4.1)
2.1Improper fractions and mixed numbers
Mixed numbers and improper fractions are two ways to write the same amount. A cake is cut into 4 equal parts. One whole cake and 3 parts of another cake is 1 3/4 cakes, or 7/4 cakes.
A mixed number has a whole number and a proper fraction, for example 1 3/4.
An improper fraction has a numerator that is larger than (or equal to) the denominator, for example 7/4.
An improper fraction has a numerator that is larger than (or equal to) the denominator, for example 7/4.
Mixed number to improper fraction
Worked example: write 3 2/5 as an improper fraction
Step 1. Multiply the whole number by the denominator: 3 × 5 = 15.Step 2. Add the numerator: 15 + 2 = 17. This is the new numerator.
Step 3. Keep the denominator: 3 2/5 = 17/5.
Improper fraction to mixed number
Worked example: write 23/6 as a mixed number
Step 1. Divide the numerator by the denominator: 23 ÷ 6 = 3 remainder 5.Step 2. The quotient is the whole number (3). The remainder is the new numerator (5). The divisor is the denominator (6).
Step 3. 23/6 = 3 5/6.
Methods follow Mathematics Year 4 (DLP), pp. 67-70.
The remainder goes on top, not next to the whole number. 23/6 is 3 5/6, not "3 and 5". Also, 15/4 is not 1 1/4: 15 ÷ 4 = 3 remainder 3, so 15/4 = 3 3/4.
Write 5 3/10 as an improper fraction, and 31/4 as a mixed number.
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5 × 10 + 3 = 53, so 5 3/10 = 53/10. 31 ÷ 4 = 7 remainder 3, so 31/4 = 7 3/4.
2.2Adding fractions
Add the numerators and keep the denominator, as long as the denominators are the same. If the answer is an improper fraction, change it to a mixed number.
Worked example: 2/9 + 4/9 + 5/9
Step 1. Same denominator (9), so add the numerators: 2 + 4 + 5 = 11.Step 2. 11/9 is an improper fraction. 11 ÷ 9 = 1 remainder 2.
Step 3. 2/9 + 4/9 + 5/9 = 1 2/9.
Adding mixed numbers
Add the whole numbers first, then the fractions. Make the denominators the same before you add. The smallest common denominator is the smallest number that both denominators divide into (a common multiple, found from the times tables).
Worked example: 1 2/3 + 2 1/2
Step 1. Whole numbers: 1 + 2 = 3.Step 2. Fractions: the smallest common denominator of 3 and 2 is 6. 2/3 = 4/6 and 1/2 = 3/6.
Step 3. 4/6 + 3/6 = 7/6 = 1 1/6.
Step 4. 3 + 1 1/6 = 4 1/6.
Never add the denominators. 2/7 + 4/7 = 6/7, not 6/14. The denominator tells the size of the parts, and the parts do not change.
Methods follow Mathematics Year 4 (DLP), pp. 71-74.
Calculate 3/8 + 1 3/4. Give the answer as a mixed number in its simplest form.
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1 3/4 = 1 6/8. 3/8 + 6/8 = 9/8 = 1 1/8. Add the whole number: 1 + 1 1/8 = 2 1/8.
2.3Subtracting fractions
The rule is the same: the denominators must match first. Subtract the numerators and keep the denominator. Then write the answer in the simplest form (divide the numerator and denominator by the same number).
Worked example: 1 7/8 − 5/8
Step 1. The denominators are the same. Subtract the numerators: 7 − 5 = 2, so the answer is 1 2/8.Step 2. Simplify 2/8 by dividing by 2: 1/4.
Step 3. 1 7/8 − 5/8 = 1 1/4.
Subtracting from a whole number, or when the fraction is too small
Worked example: 4 − 2 3/4 and 5 1/4 − 2 3/4
4 − 2 3/4. Take 1 from the 4 and write it as 4/4. So 4 = 3 4/4. Then 3 4/4 − 2 3/4 = 1 1/4.5 1/4 − 2 3/4. 1/4 is smaller than 3/4, so regroup: 5 1/4 = 4 5/4. Then 4 5/4 − 2 3/4 = 2 2/4 = 2 1/2.
Not sure? Change both numbers to improper fractions first. 5 1/4 = 21/4 and 2 3/4 = 11/4. Then 21/4 − 11/4 = 10/4 = 2 2/4 = 2 1/2. You get the same answer.
Different denominators
Worked example: 4 1/2 − 1 1/3
Step 1. The smallest common denominator of 2 and 3 is 6. 1/2 = 3/6 and 1/3 = 2/6.Step 2. Whole numbers: 4 − 1 = 3. Fractions: 3/6 − 2/6 = 1/6.
Step 3. 4 1/2 − 1 1/3 = 3 1/6.
Methods follow Mathematics Year 4 (DLP), pp. 75-79.
Calculate 6 1/5 − 2 1/2.
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Common denominator 10: 6 2/10 − 2 5/10. 2/10 is smaller than 5/10, so regroup: 5 12/10 − 2 5/10 = 3 7/10. The answer is 3 7/10.
2.4Adding and subtracting in one sum
Work from left to right. Do the first operation, then the second. Make the denominators the same before you start.
Worked example: 3/4 + 5/8 − 1/2
Step 1. The smallest common denominator of 4, 8 and 2 is 8: 3/4 = 6/8 and 1/2 = 4/8.Step 2. Left to right: 6/8 + 5/8 = 11/8.
Step 3. 11/8 − 4/8 = 7/8. It is already in the simplest form.
Do not subtract first just because it looks easier. 9/10 − 3/10 + 4/10 = 6/10 + 4/10 = 1. If you did 3/10 + 4/10 first and then 9/10 − 7/10, you would get 2/10, which is wrong.
Methods follow Mathematics Year 4 (DLP), pp. 80-82.
Calculate 7/10 + 3/5 − 1/2.
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3/5 = 6/10 and 1/2 = 5/10. 7/10 + 6/10 = 13/10. 13/10 − 5/10 = 8/10 = 4/5.
2.5Fractions of a quantity
"Of" means multiply. To find 3/8 of 40, work out 3/8 × 40.
Worked example: 5/7 of 35 pieces of penganan
Step 1. Divide 35 into 7 equal parts: 35 ÷ 7 = 5 in each part.Step 2. Take 5 parts: 5 × 5 = 25.
Step 3. 5/7 of 35 = 25.
Worked example: 2 1/2 of 30 metres of ribbon
Step 1. 2 × 30 = 60.Step 2. 1/2 of 30 = 15.
Step 3. 60 + 15 = 75 m. (Or change 2 1/2 to 5/2: 5 × 30 ÷ 2 = 75.)
Divide by the denominator first, then multiply by the numerator. That keeps the numbers small and you avoid mistakes.
Methods follow Mathematics Year 4 (DLP), pp. 83-85.
Find 3/8 of 64 kg, and 1 2/5 of 45 litres.
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64 ÷ 8 = 8, 8 × 3 = 24 kg. 45 ÷ 5 = 9, so 1/5 = 9 and 2/5 = 18. 45 + 18 = 63 litres.
2.6Adding and subtracting decimals
Decimals up to three decimal places have tenths, hundredths and thousandths. Add and subtract them like whole numbers, but line up the decimal points.
| Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|
| 4 | . | 2 | 0 | 7 |
Write 0 as a placeholder so that every number has the same number of decimal places.
Worked example: 1.3 kg + 4.207 kg + 0.96 kg
Step 1. Line up the decimal points and add 0s: 1.300, 4.207 and 0.960.Step 2. Add as whole numbers: 1 300 + 4 207 + 960 = 6 467.
Step 3. Put the decimal point in the same column: 6.467 kg.
Worked example: 10 − 3.462
Step 1. Write 10 as 10.000 so the places match.Step 2. 10 000 − 3 462 = 6 538.
Step 3. 10 − 3.462 = 6.538. Check by adding: 6.538 + 3.462 = 10.000.
Do not line up the last digits. In 7.918 − 3, the 3 is 3.000, so the answer is 4.918, not 7.915.
Methods follow Mathematics Year 4 (DLP), pp. 86-89.
Calculate 12.4 + 0.375 and 5.2 − 1.86.
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12.400 + 0.375 = 12.775. 5.20 − 1.86 = 3.34.
2.7Multiplying and dividing decimals
By a whole number
Multiply or divide as if there is no decimal point. Then put the point back so the answer has the same number of decimal places as the decimal number.
Worked example: 4 × 2.54 cm
Step 1. 4 × 254 = 1 016.Step 2. 2.54 has 2 decimal places, so the answer has 2 decimal places.
Step 3. 4 × 2.54 = 10.16 cm.
Worked example: 14.96 m ÷ 8
Step 1. Divide as whole numbers: 1 496 ÷ 8 = 187.Step 2. Place the decimal point so there are 2 decimal places: 1.87.
Step 3. 14.96 ÷ 8 = 1.87 m. Check: 1.87 × 8 = 14.96.
By 10, 100 and 1 000
| Operation | What happens | Example |
|---|---|---|
| × 10 | point moves 1 place right | 10 × 1.25 = 12.5 |
| × 100 | point moves 2 places right | 100 × 7.342 = 734.2 |
| × 1 000 | point moves 3 places right | 1 000 × 0.956 = 956 |
| ÷ 10 | point moves 1 place left | 4.32 ÷ 10 = 0.432 |
| ÷ 100 | point moves 2 places left | 657.8 ÷ 100 = 6.578 |
| ÷ 1 000 | point moves 3 places left | 2 749 ÷ 1 000 = 2.749 |
When a zero is the last decimal digit, you can drop it: 12.50 = 12.5. But a zero in the middle must stay: 12.05 is not 12.5.
Estimate first by rounding. 3 × 1.7 is about 3 × 2 = 6, so 5.1 is a sensible answer, and 51 is not.
Methods follow Mathematics Year 4 (DLP), pp. 90-93.
Calculate 6 × 3.15, 7.2 ÷ 8 and 0.45 × 100.
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6 × 315 = 1 890, so 18.9 (2 decimal places: 18.90). 72 ÷ 8 = 9, so 7.2 ÷ 8 = 0.9. 0.45 × 100 = 45.
2.8Fractions and percentages
Percentage means "out of 100". The symbol % is read "per cent". 35% = 35/100 = 35 hundredths.
Fraction to percentage
Method 1. Make the denominator 100. 4/5 = (4 × 20)/(5 × 20) = 80/100 = 80%.Method 2. Multiply by 100%. 4/5 × 100% = 4 × 20% = 80%.
Percentage to fraction
Write the percentage over 100, then simplify. 40% = 40/100 = 40 ÷ 20 over 100 ÷ 20 = 2/5.Percentage of objects: 12 out of 50 books
Write the fraction 12/50. Make the denominator 100: (12 × 2)/(50 × 2) = 24/100 = 24%.Denominators 2, 4, 5, 10, 20, 25 and 50 are easy: they all go into 100. 1/4 = 25%, 1/2 = 50%, 3/4 = 75%, 1/5 = 20%.
A percentage is out of 100, not out of the total in the question. 8 vans out of 40 vehicles is 20%, not 8%.
Methods follow Mathematics Year 4 (DLP), pp. 94-96.
Write 7/20 as a percentage and 45% as a fraction in the simplest form. What is the percentage of 18 out of 25?
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7/20 = 35/100 = 35%. 45% = 45/100 = 9/20. 18/25 = 72/100 = 72%.
2.9Solving problems
Use four steps: understand the problem, plan the strategy, solve, then check.
Worked example
A club has 140 members. 3/7 are Year 4 pupils and 1/2 are Year 5 pupils. What is the difference between the numbers of Year 4 and Year 5 pupils?Step 1. Year 4: 3/7 × 140 = 20 × 3 = 60.
Step 2. Year 5: 1/2 × 140 = 70.
Step 3. Difference: 70 − 60 = 10.
Step 4. Check: 60 + 10 = 70.
Worked example: percentage spent
RM70 of a RM280 fund was spent. 70/280 = 1/4 = 25/100, so 25% was spent. Check: 25% of 280 = 25/100 × 280 = 70.Underline the important numbers and key words. "Total" means add, "difference" means subtract, "each" often means divide. Draw a bar model when the question has fractions of a total.
Methods follow Mathematics Year 4 (DLP), pp. 97-104.
Mei Ling has a 3.6 m ribbon. She uses 1/4 of it. How many metres are left?
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1/4 of 3.6 = 3.6 ÷ 4 = 0.9 m used. 3.6 − 0.9 = 2.7 m left.