GCSE Maths: Fractions, decimals and percentages (Foundation)
Part of the GCSE Maths fractions chapter (Foundation). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Fractions, decimals and percentages
Fractions, decimals and percentages are three ways to write the same number. These are the equivalents worth knowing by heart.
| Fraction | Decimal | Percentage |
|---|---|---|
| 12 | 0.5 | 50% |
| 14 | 0.25 | 25% |
| 34 | 0.75 | 75% |
| 15 | 0.2 | 20% |
| 110 | 0.1 | 10% |
| 18 | 0.125 | 12.5% |
| 13 | 0.333… (0.3) | 33.3…% |
Key ideas
- Fraction → decimal: divide the numerator by the denominator.
- Decimal → fraction: use place value (tenths, hundredths, thousandths), then simplify.
- Fraction → percentage: make the denominator 100 (or convert to a decimal and multiply by 100).
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.38 = 3 ÷ 8
- Set up the division 3 ÷ 8. 3 is smaller than 8, so the answer is less than 1. Write 3 as 3.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.3.000… ÷ 8
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.8 into 3 goes 0 times, remainder 3
8 into 30 goes 3 times, remainder 6
8 into 60 goes 7 times, remainder 4
8 into 40 goes 5 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.3 ÷ 8 = 0.375
Check. Multiply back: 0.375 × 8 = 3, which equals 3. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.65 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.65 = 65100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 65: 1, 5, 13, 65
HCF of 65 and 100 = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value and gives smaller numbers.65 ÷ 5 = 13 and 100 ÷ 5 = 20
65100 = 1320
Check. Divide the top by the bottom: 1320 gives 65 ÷ 100 = 0.65. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.35% = 35 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.7 ÷ 20 = 0.35
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.35 × 100 = 35
- Write the percentage. Add the % sign.720 = 35%
Check. A different route: 720 = 35100 (multiply top and bottom by 5), which is 35%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.04 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.04 = 4100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 4: 1, 2, 4
HCF of 4 and 100 = 4 - Divide the top and the bottom by 4. Dividing both by the same number keeps the value and gives smaller numbers.4 ÷ 4 = 1 and 100 ÷ 4 = 25
4100 = 125
Check. Divide the top by the bottom: 125 gives 4 ÷ 100 = 0.04. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.22.5% = 22.5 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.9 ÷ 40 = 0.225
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.225 × 100 = 22.5
- Write the percentage. Add the % sign.940 = 22.5%
Check. A different route: 22.5% of 100 is 22.5, and 940 of 100 = 100 ÷ 40 × 9 = 2.5 × 9 = 22.5. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To compare numbers in different forms, change them all to the same form. Percentages are often the easiest because they are all out of 100.
- Choose one form. We have a fraction, a decimal and a percentage. Percentages are all “out of 100”, so they are easy to compare.Change everything to a percentage.
- Change 34 to a percentage. Multiply the top and the bottom by 25 to make the denominator 100.34 = 75100 = 75%
- Change 0.7 to a percentage. Multiply a decimal by 100 to get a percentage.0.7 × 100 = 70%
- Keep 68% as it is. It is already a percentage.68%
- Put the percentages in order. Smallest first: compare 75, 70 and 68.68%, 70%, 75%
- Write the original numbers in that order. The question asks for the original values, not the percentages.68%, 0.7, 34
Check. A different route: change everything to decimals. 68% = 0.68, 0.7 stays 0.7, 34 = 0.75. Order: 0.68, 0.7, 0.75. ✓
Watch out. 0.7 is 70%, not 7%. Line the decimals up by place value before comparing.
Exercise 3I Fractions, decimals and percentages
Non-calculatorMethod. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.12 = 1 ÷ 2
- Set up the division 1 ÷ 2. 1 is smaller than 2, so the answer is less than 1. Write 1 as 1.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.1.000… ÷ 2
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.2 into 1 goes 0 times, remainder 1
2 into 10 goes 5 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.1 ÷ 2 = 0.5
Check. Multiply back: 0.5 × 2 = 1, which equals 1. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.34 = 3 ÷ 4
- Set up the division 3 ÷ 4. 3 is smaller than 4, so the answer is less than 1. Write 3 as 3.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.3.000… ÷ 4
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.4 into 3 goes 0 times, remainder 3
4 into 30 goes 7 times, remainder 2
4 into 20 goes 5 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.3 ÷ 4 = 0.75
Check. Multiply back: 0.75 × 4 = 3, which equals 3. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.15 = 1 ÷ 5
- Set up the division 1 ÷ 5. 1 is smaller than 5, so the answer is less than 1. Write 1 as 1.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.1.000… ÷ 5
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.5 into 1 goes 0 times, remainder 1
5 into 10 goes 2 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.1 ÷ 5 = 0.2
Check. Multiply back: 0.2 × 5 = 1, which equals 1. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.710 = 7 ÷ 10
- Set up the division 7 ÷ 10. 7 is smaller than 10, so the answer is less than 1. Write 7 as 7.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.7.000… ÷ 10
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.10 into 7 goes 0 times, remainder 7
10 into 70 goes 7 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.7 ÷ 10 = 0.7
Check. Multiply back: 0.7 × 10 = 7, which equals 7. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.38 = 3 ÷ 8
- Set up the division 3 ÷ 8. 3 is smaller than 8, so the answer is less than 1. Write 3 as 3.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.3.000… ÷ 8
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.8 into 3 goes 0 times, remainder 3
8 into 30 goes 3 times, remainder 6
8 into 60 goes 7 times, remainder 4
8 into 40 goes 5 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.3 ÷ 8 = 0.375
Check. Multiply back: 0.375 × 8 = 3, which equals 3. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. A fraction bar means divide: the numerator is divided by the denominator.
- Say what the fraction means. The fraction bar means "divided by", so the top number is divided by the bottom number.920 = 9 ÷ 20
- Set up the division 9 ÷ 20. 9 is smaller than 20, so the answer is less than 1. Write 9 as 9.000… (adding zeros after the decimal point does not change its value) so we can keep dividing.9.000… ÷ 20
- Divide step by step (short division). At each step carry the remainder across as the next digit and put a zero after it. Stop when the remainder is 0.20 into 9 goes 0 times, remainder 9
20 into 90 goes 4 times, remainder 10
20 into 100 goes 5 times, remainder 0 - Write the digits as a decimal. Put the decimal point after the first digit, then the digits found in order.9 ÷ 20 = 0.45
Check. Multiply back: 0.45 × 20 = 9, which equals 9. ✓
Watch out. Divide the top by the bottom, not the bottom by the top. The answer to a proper fraction is always less than 1.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.5 has 1 decimal place, so it is in tenths.
- Write the digits as a fraction over 10, 100 or 1000. 1 decimal place means the denominator is 1 followed by 1 zero.0.5 = 510
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 5: 1, 5
HCF of 5 and 10 = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value and gives smaller numbers.5 ÷ 5 = 1 and 10 ÷ 5 = 2
510 = 12
Check. Divide the top by the bottom: 12 gives 5 ÷ 10 = 0.5. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.2 has 1 decimal place, so it is in tenths.
- Write the digits as a fraction over 10, 100 or 1000. 1 decimal place means the denominator is 1 followed by 1 zero.0.2 = 210
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 2: 1, 2
HCF of 2 and 10 = 2 - Divide the top and the bottom by 2. Dividing both by the same number keeps the value and gives smaller numbers.2 ÷ 2 = 1 and 10 ÷ 2 = 5
210 = 15
Check. Divide the top by the bottom: 15 gives 2 ÷ 10 = 0.2. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.75 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.75 = 75100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 75: 1, 3, 5, 15, 25, 75
HCF of 75 and 100 = 25 - Divide the top and the bottom by 25. Dividing both by the same number keeps the value and gives smaller numbers.75 ÷ 25 = 3 and 100 ÷ 25 = 4
75100 = 34
Check. Divide the top by the bottom: 34 gives 75 ÷ 100 = 0.75. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.35 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.35 = 35100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 35: 1, 5, 7, 35
HCF of 35 and 100 = 5 - Divide the top and the bottom by 5. Dividing both by the same number keeps the value and gives smaller numbers.35 ÷ 5 = 7 and 100 ÷ 5 = 20
35100 = 720
Check. Divide the top by the bottom: 720 gives 35 ÷ 100 = 0.35. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.08 has 2 decimal places, so it is in hundredths.
- Write the digits as a fraction over 10, 100 or 1000. 2 decimal places means the denominator is 1 followed by 2 zeros.0.08 = 8100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 8: 1, 2, 4, 8
HCF of 8 and 100 = 4 - Divide the top and the bottom by 4. Dividing both by the same number keeps the value and gives smaller numbers.8 ÷ 4 = 2 and 100 ÷ 4 = 25
8100 = 225
Check. Divide the top by the bottom: 225 gives 8 ÷ 100 = 0.08. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.125 has 3 decimal places, so it is in thousandths.
- Write the digits as a fraction over 10, 100 or 1000. 3 decimal places means the denominator is 1 followed by 3 zeros.0.125 = 1251000
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 125: 1, 5, 25, 125
HCF of 125 and 1000 = 125 - Divide the top and the bottom by 125. Dividing both by the same number keeps the value and gives smaller numbers.125 ÷ 125 = 1 and 1000 ÷ 125 = 8
1251000 = 18
Check. Divide the top by the bottom: 18 gives 125 ÷ 1000 = 0.125. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.25% = 25 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.1 ÷ 4 = 0.25
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.25 × 100 = 25
- Write the percentage. Add the % sign.14 = 25%
Check. A different route: 14 = 25100 (multiply top and bottom by 25), which is 25%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.60% = 60 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.3 ÷ 5 = 0.6
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.6 × 100 = 60
- Write the percentage. Add the % sign.35 = 60%
Check. A different route: 35 = 60100 (multiply top and bottom by 20), which is 60%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.35% = 35 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.7 ÷ 20 = 0.35
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.35 × 100 = 35
- Write the percentage. Add the % sign.720 = 35%
Check. A different route: 720 = 35100 (multiply top and bottom by 5), which is 35%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.90% = 90 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.9 ÷ 10 = 0.9
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.9 × 100 = 90
- Write the percentage. Add the % sign.910 = 90%
Check. A different route: 910 = 90100 (multiply top and bottom by 10), which is 90%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.12% = 12 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.3 ÷ 25 = 0.12
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.12 × 100 = 12
- Write the percentage. Add the % sign.325 = 12%
Check. A different route: 325 = 12100 (multiply top and bottom by 4), which is 12%. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. To change a fraction to a percentage: change to a decimal, then multiply by 100.
- Say what percent means. Per cent means "out of 100". So a percentage is just a fraction with denominator 100.62.5% = 62.5 out of 100
- Change the fraction to a decimal. Divide the top by the bottom.5 ÷ 8 = 0.625
- Multiply by 100. Multiplying by 100 turns "out of 1" into "out of 100".0.625 × 100 = 62.5
- Write the percentage. Add the % sign.58 = 62.5%
Check. A different route: 62.5% of 100 is 62.5, and 58 of 100 = 100 ÷ 8 × 5 = 12.5 × 5 = 62.5. ✓
Watch out. Do not forget the % sign, and remember it is multiply by 100 (not divide).
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.20% = 20100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 20 - Divide the top and the bottom by 20. The same number is divided into both, so the value does not change.20 ÷ 20 = 1 and 100 ÷ 20 = 5
20100 = 15
Check. 15 = 0.2 and 20% = 0.2. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.45% = 45100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 5 - Divide the top and the bottom by 5. The same number is divided into both, so the value does not change.45 ÷ 5 = 9 and 100 ÷ 5 = 20
45100 = 920
Check. 920 = 0.45 and 45% = 0.45. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.12% = 12100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 4 - Divide the top and the bottom by 4. The same number is divided into both, so the value does not change.12 ÷ 4 = 3 and 100 ÷ 4 = 25
12100 = 325
Check. 325 = 0.12 and 12% = 0.12. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.75% = 75100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 75: 1, 3, 5, 15, 25, 75
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 25 - Divide the top and the bottom by 25. The same number is divided into both, so the value does not change.75 ÷ 25 = 3 and 100 ÷ 25 = 4
75100 = 34
Check. 34 = 0.75 and 75% = 0.75. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.5% = 5100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 5: 1, 5
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 5 - Divide the top and the bottom by 5. The same number is divided into both, so the value does not change.5 ÷ 5 = 1 and 100 ÷ 5 = 20
5100 = 120
Check. 120 = 0.05 and 5% = 0.05. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
Method. A percentage is a fraction out of 100: write it over 100, then simplify.
- Say what percent means. Per cent means "out of 100", so the percentage goes on the top of a fraction with 100 on the bottom.60% = 60100
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 20 - Divide the top and the bottom by 20. The same number is divided into both, so the value does not change.60 ÷ 20 = 3 and 100 ÷ 20 = 5
60100 = 35
Check. 35 = 0.6 and 60% = 0.6. They match. ✓
Watch out. The 100 stays on the bottom until you simplify. 45% is not 45 over 45.
| Fraction | Decimal | Percentage |
|---|---|---|
| 35 | ? | ? |
| ? | 0.4 | ? |
| ? | ? | 85% |
Method. Convert between the three forms: fraction → decimal by dividing, decimal → fraction by place value, and decimal → percentage by multiplying by 100.
- Row 1: fraction to decimal. The fraction bar means divide: 3 ÷ 5.35 = 3 ÷ 5 = 0.6
- Row 1: decimal to percentage. Multiply by 100 (percent means out of 100).0.6 × 100 = 60%
- Row 2: decimal to fraction. 0.4 has one decimal place, so it is tenths. Then simplify by dividing the top and bottom by 2.0.4 = 410 = 25
- Row 2: decimal to percentage. Multiply by 100.0.4 × 100 = 40%
- Row 3: percentage to fraction. Percent means out of 100, so write 85 over 100, then simplify by dividing top and bottom by 5.85% = 85100 = 1720
- Row 3: percentage to decimal. Divide by 100.85 ÷ 100 = 0.85
Check. Check each row says the same number in three ways: 35 = 0.6 = 60%, 25 = 0.4 = 40%, 1720 = 0.85 = 85%. ✓
Watch out. A percentage is always out of 100. Do not mix up the divide-by-100 and multiply-by-100 steps.
Method. To compare numbers in mixed forms, change everything to the same form. Percentages are easiest here.
- Choose one form. Percentages are all out of 100, so they are easy to compare.Change everything to a percentage.
- Change 35 to a percentage. Multiply the top and the bottom by 20 to make the denominator 100.35 = 60100 = 60%
- Change 0.55 to a percentage. Multiply by 100.0.55 × 100 = 55%
- Change 58 to a percentage. Divide 5 by 8, then multiply by 100.5 ÷ 8 = 0.625
0.625 × 100 = 62.5% - Put the percentages in order. 58% needs no change. Smallest first: 55, 58, 60, 62.5.55%, 58%, 60%, 62.5%
- Write the original values in that order. The question asks for the original values.0.55, 58%, 35, 58
Check. A different route: change to decimals. 35 = 0.6, 0.55, 58% = 0.58, 58 = 0.625. Order: 0.55, 0.58, 0.6, 0.625. Same order. ✓
Watch out. Compare the decimals digit by digit: 0.55 < 0.58 < 0.6 < 0.625.
Method. Convert both scores to percentages to compare them.
- Choose the form. Raj's score is a percentage, so change Priya's fraction into a percentage as well.Priya 1720, Raj 82%
- Change Priya's score to a percentage. Multiply the top and the bottom by 5 to make the denominator 100.1720 = 85100 = 85%
- Compare. Raj scored 82%.85% > 82%
- Write the conclusion. The bigger score is the better score.Priya did better.
Check. A different route: change Raj's score to a fraction over 20 as a decimal. 82% = 0.82, and 1720 = 0.85. 0.85 > 0.82. ✓
Watch out. Compare numbers only after writing them in the same form.
Method. Divide 1 by 3 and look at the decimal. Then compare it with 0.3.
- Divide 1 by 3. The fraction bar means divide. Add zeros and divide step by step.3 into 1 = 0 remainder 1
3 into 10 = 3 remainder 1
3 into 10 = 3 remainder 1
The same remainder keeps coming back - Say what this means. The remainder is 1 every time, so the digit 3 repeats for ever.1 ÷ 3 = 0.3333…
- Compare with 0.3. 0.3 is exactly three tenths. Use a common denominator of 30 to compare with one third.0.3 = 310 = 930
13 = 1030 - Conclude. 9 is smaller than 10, so 0.3 is smaller than one third. Sara is not quite right.1/3 = 0.3 (recurring), which is slightly more than 0.3
Check. Multiply 0.3 by 3: 0.3 × 3 = 0.9, not 1. But one third × 3 = 1. So 0.3 cannot be exactly one third. ✓
Watch out. A recurring decimal never ends. Writing it as 0.3 is only an approximation.
Method. Use place value to write the decimal as a fraction over 10, 100 or 1000, then simplify.
- Decide what is asked. To write a mass as a fraction of 1 kg we write the decimal as a fraction of 1.0.375 kg out of 1 kg
- Count the digits after the decimal point. Each digit is a place value: the first is tenths, the second is hundredths, the third is thousandths. This tells us the denominator.0.375 has 3 decimal places, so it is in thousandths.
- Write the digits as a fraction over 10, 100 or 1000. 3 decimal places means the denominator is 1 followed by 3 zeros.0.375 = 3751000
- Find the highest common factor (HCF) of the top and the bottom. We want to simplify the fraction fully.Factors of 375: 1, 3, 5, 15, 25, 75, 125, 375
HCF of 375 and 1000 = 125 - Divide the top and the bottom by 125. Dividing both by the same number keeps the value and gives smaller numbers.375 ÷ 125 = 3 and 1000 ÷ 125 = 8
3751000 = 38 - Write the answer with its units. The fraction is a fraction of one kilogram.38 kg
Check. Divide the top by the bottom: 38 gives 375 ÷ 1000 = 0.375. ✓
Watch out. Count the decimal places carefully: 0.08 is hundredths (8 over 100), not tenths.