GCSE Maths: Equivalent fractions and simplifying (Foundation)
Part of the GCSE Maths fractions chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Equivalent fractions and simplifying
Equivalent fractions are different-looking fractions that have the same value. Look at the bars: they have the same amount shaded.
Key ideas
- To make an equivalent fraction, multiply or divide the numerator and the denominator by the same number.
- To simplify a fraction, divide the top and bottom by a common factor. A fraction is in its simplest form when no number (other than 1) divides into both.
- The quickest way: divide by the highest common factor (HCF) of the numerator and denominator.
Method. Equivalent fractions: whatever you multiply the bottom by, multiply the top by the same number.
- Read the question. We have a fraction with a missing top number. The new bottom number is given, so we must find out how the bottom changed.35 = ?20
- Find the multiplier for the bottom: 5 → 20. Ask: what do we multiply the old bottom by to get the new bottom? Divide the new bottom by the old bottom.20 ÷ 5 = 4, so the bottom was multiplied by 4.
- Multiply the top by 4 as well. To keep the fraction the same size we must multiply the top and the bottom by the same number.3 × 4 = 12
- Write the full equation. Now put the new top into the gap.35 = 1220
Check. Divide the top and bottom of 1220 by 4 and we get 35 back. Also 3 × 20 = 60 and 12 × 5 = 60, which are equal. ✓
Watch out. Never add the same number to the top and the bottom. You must multiply (or divide).
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.1830
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 3, 6
HCF = 6 - Divide the top by 6. Whatever we do to the top we must also do to the bottom.18 ÷ 6 = 3
- Divide the bottom by 6. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.30 ÷ 6 = 5
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.1830 = 35
Check. A different route: divide in small steps (÷ 2, then ÷ 3). 1830 → 915 → 35. We reach the same fraction. As decimals, 1830 = 0.6 and 35 = 0.6. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Two fractions are equivalent if they have the same value. Simplify both fully: if they reach the same fraction, they are equivalent.
- Say what we need to show. Equivalent fractions look different but have the same value. The clearest test is to reduce each one to its simplest form and compare.46 and 1015
- Simplify 46. The biggest number that divides into both 4 and 6 is 2 (factors of 4: 1, 2, 4. Factors of 6: 1, 2, 3, 6). Divide the top and the bottom by it.4 ÷ 2 = 2 and 6 ÷ 2 = 3
46 = 23 - Simplify 1015. The biggest number that divides into both 10 and 15 is 5 (factors of 10: 1, 2, 5, 10. Factors of 15: 1, 3, 5, 15). Divide the top and the bottom by it.10 ÷ 5 = 2 and 15 ÷ 5 = 3
1015 = 23 - Compare the simplest forms. Both fractions have reduced to the same fraction, so they have the same value.23 = 23
- Write the conclusion. Answer the question that was asked.Yes, 46 and 1015 are equivalent.
Check. A different test: cross-multiply. 4 × 15 = 60 and 6 × 10 = 60. The two answers are equal, so the fractions are equivalent. Also both equal 0.667 as a decimal. ✓
Watch out. Fractions with different numbers can still be the same fraction. Always simplify before you decide.
Method. If the highest common factor is hard to spot, divide the top and the bottom by any common factor you can see, then repeat until no number except 1 divides into both.
- Look for an easy common factor. Both 84 and 126 are even, so both are in the 2 times table. We can always divide two even numbers by 2.84 ÷ 2 = 42 and 126 ÷ 2 = 63
84126 = 4263 - Look at the new fraction and divide again. 42 and 63 are both in the 3 times table (the digits of 42 add to 6 and the digits of 63 add to 9, and both totals are in the 3 times table).42 ÷ 3 = 14 and 63 ÷ 3 = 21
4263 = 1421 - Divide again. 14 and 21 are both in the 7 times table.14 ÷ 7 = 2 and 21 ÷ 7 = 3
1421 = 23 - Stop when nothing is left to divide out. 2 and 3 have no common factor except 1, so the fraction cannot be simplified any more.84126 = 23
Check. The three divisions were by 2, 3 and 7, and 2 × 3 × 7 = 42. So 42 is the highest common factor. One division by 42 gives 84 ÷ 42 = 2 and 126 ÷ 42 = 3, the same answer 23. ✓
Watch out. Do not stop too soon. 4263 is simpler than 84126 but it is not yet in its simplest form.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. £2.40 is 240 pence. A fraction compares two amounts, so they must be in the same units. Change the pounds to pence.Part = 36p, whole = 240p
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).36240
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
HCF = 12 - Divide the top and the bottom by 12. Dividing both by the same number keeps the value the same but gives smaller numbers.36 ÷ 12 = 3 and 240 ÷ 12 = 20
36240 = 320
Check. 36 ÷ 240 = 0.15 and 320 = 0.15. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Watch out
Adding the same number to the top and bottom does not make an equivalent fraction. 12 → 34 = 34 changes the value. You must multiply or divide.Exercise 3B Equivalent fractions and simplifying
Non-calculatorMethod. Equivalent fractions: whatever you multiply the bottom by, multiply the top by the same number.
- Read the question. We have a fraction with a missing top number. The new bottom number is given, so we must find out how the bottom changed.12 = ?6
- Find the multiplier for the bottom: 2 → 6. Ask: what do we multiply the old bottom by to get the new bottom? Divide the new bottom by the old bottom.6 ÷ 2 = 3, so the bottom was multiplied by 3.
- Multiply the top by 3 as well. To keep the fraction the same size we must multiply the top and the bottom by the same number.1 × 3 = 3
- Write the full equation. Now put the new top into the gap.12 = 36
Check. Divide the top and bottom of 36 by 3 and we get 12 back. Also 1 × 6 = 6 and 3 × 2 = 6, which are equal. ✓
Watch out. Never add the same number to the top and the bottom. You must multiply (or divide).
Method. Equivalent fractions: whatever you multiply the bottom by, multiply the top by the same number.
- Read the question. We have a fraction with a missing top number. The new bottom number is given, so we must find out how the bottom changed.13 = ?12
- Find the multiplier for the bottom: 3 → 12. Ask: what do we multiply the old bottom by to get the new bottom? Divide the new bottom by the old bottom.12 ÷ 3 = 4, so the bottom was multiplied by 4.
- Multiply the top by 4 as well. To keep the fraction the same size we must multiply the top and the bottom by the same number.1 × 4 = 4
- Write the full equation. Now put the new top into the gap.13 = 412
Check. Divide the top and bottom of 412 by 4 and we get 13 back. Also 1 × 12 = 12 and 4 × 3 = 12, which are equal. ✓
Watch out. Never add the same number to the top and the bottom. You must multiply (or divide).
Method. Equivalent fractions: whatever you multiply the bottom by, multiply the top by the same number.
- Read the question. We have a fraction with a missing top number. The new bottom number is given, so we must find out how the bottom changed.25 = ?15
- Find the multiplier for the bottom: 5 → 15. Ask: what do we multiply the old bottom by to get the new bottom? Divide the new bottom by the old bottom.15 ÷ 5 = 3, so the bottom was multiplied by 3.
- Multiply the top by 3 as well. To keep the fraction the same size we must multiply the top and the bottom by the same number.2 × 3 = 6
- Write the full equation. Now put the new top into the gap.25 = 615
Check. Divide the top and bottom of 615 by 3 and we get 25 back. Also 2 × 15 = 30 and 6 × 5 = 30, which are equal. ✓
Watch out. Never add the same number to the top and the bottom. You must multiply (or divide).
Method. Equivalent fractions: whatever you multiply the bottom by, multiply the top by the same number.
- Read the question. We have a fraction with a missing top number. The new bottom number is given, so we must find out how the bottom changed.34 = ?20
- Find the multiplier for the bottom: 4 → 20. Ask: what do we multiply the old bottom by to get the new bottom? Divide the new bottom by the old bottom.20 ÷ 4 = 5, so the bottom was multiplied by 5.
- Multiply the top by 5 as well. To keep the fraction the same size we must multiply the top and the bottom by the same number.3 × 5 = 15
- Write the full equation. Now put the new top into the gap.34 = 1520
Check. Divide the top and bottom of 1520 by 5 and we get 34 back. Also 3 × 20 = 60 and 15 × 4 = 60, which are equal. ✓
Watch out. Never add the same number to the top and the bottom. You must multiply (or divide).
Method. Equivalent fractions: whatever you multiply the top by, multiply the bottom by the same number.
- Read the question. We have a fraction with a missing bottom number. The new top number is given, so we must find how the top changed.78 = 21?
- Find the multiplier for the top: 7 → 21. Divide the new top by the old top to see what the top was multiplied by.21 ÷ 7 = 3, so the top was multiplied by 3.
- Multiply the bottom by 3 as well. Top and bottom must be scaled by the same number or the fraction changes.8 × 3 = 24
- Write the full equation. Put the new bottom into the gap.78 = 2124
Check. As decimals, both are 0.875. Also 7 × 24 = 168 and 21 × 8 = 168, which are equal. ✓
Watch out. The same multiplier must be used on the top and the bottom.
Method. Equivalent fractions: whatever you multiply the top by, multiply the bottom by the same number.
- Read the question. We have a fraction with a missing bottom number. The new top number is given, so we must find how the top changed.49 = 20?
- Find the multiplier for the top: 4 → 20. Divide the new top by the old top to see what the top was multiplied by.20 ÷ 4 = 5, so the top was multiplied by 5.
- Multiply the bottom by 5 as well. Top and bottom must be scaled by the same number or the fraction changes.9 × 5 = 45
- Write the full equation. Put the new bottom into the gap.49 = 2045
Check. As decimals, both are 0.4444. Also 4 × 45 = 180 and 20 × 9 = 180, which are equal. ✓
Watch out. The same multiplier must be used on the top and the bottom.
Method. Equivalent fractions: whatever you multiply the top by, multiply the bottom by the same number.
- Read the question. We have a fraction with a missing bottom number. The new top number is given, so we must find how the top changed.56 = 20?
- Find the multiplier for the top: 5 → 20. Divide the new top by the old top to see what the top was multiplied by.20 ÷ 5 = 4, so the top was multiplied by 4.
- Multiply the bottom by 4 as well. Top and bottom must be scaled by the same number or the fraction changes.6 × 4 = 24
- Write the full equation. Put the new bottom into the gap.56 = 2024
Check. As decimals, both are 0.8333. Also 5 × 24 = 120 and 20 × 6 = 120, which are equal. ✓
Watch out. The same multiplier must be used on the top and the bottom.
Method. Equivalent fractions: whatever you multiply the top by, multiply the bottom by the same number.
- Read the question. We have a fraction with a missing bottom number. The new top number is given, so we must find how the top changed.27 = 12?
- Find the multiplier for the top: 2 → 12. Divide the new top by the old top to see what the top was multiplied by.12 ÷ 2 = 6, so the top was multiplied by 6.
- Multiply the bottom by 6 as well. Top and bottom must be scaled by the same number or the fraction changes.7 × 6 = 42
- Write the full equation. Put the new bottom into the gap.27 = 1242
Check. As decimals, both are 0.2857. Also 2 × 42 = 84 and 12 × 7 = 84, which are equal. ✓
Watch out. The same multiplier must be used on the top and the bottom.
Method. To make equivalent fractions, multiply the top and the bottom by the same number.
- Choose multipliers. Any whole number bigger than 1 works. We use 2, 3 and 4.Multipliers: 2, 3, 4
- Multiply the top and the bottom by 2. We must multiply both numbers by the same multiplier.14 = 28 (1 × 2 = 2, 4 × 2 = 8)
- Multiply the top and the bottom by 3. Same idea with a different multiplier.14 = 312 (1 × 3 = 3, 4 × 3 = 12)
- Multiply the top and the bottom by 4. And once more.14 = 416 (1 × 4 = 4, 4 × 4 = 16)
Check. Each fraction is 0.25 as a decimal. Also, simplifying 28, 312 and 416 (dividing by 2, 3 and 4) gives 14 each time. ✓
Watch out. Multiply, do not add. 1/4 is not equal to 3/6 (adding 2 to each number).
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.48
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 4: 1, 2, 4
Factors of 8: 1, 2, 4, 8 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 4
HCF = 4 - Divide the top by 4. Whatever we do to the top we must also do to the bottom.4 ÷ 4 = 1
- Divide the bottom by 4. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.8 ÷ 4 = 2
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.48 = 12
Check. A different route: divide in small steps (÷ 2, then ÷ 2). 48 → 24 → 12. We reach the same fraction. As decimals, 48 = 0.5 and 12 = 0.5. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.69
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 6: 1, 2, 3, 6
Factors of 9: 1, 3, 9 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 3
HCF = 3 - Divide the top by 3. Whatever we do to the top we must also do to the bottom.6 ÷ 3 = 2
- Divide the bottom by 3. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.9 ÷ 3 = 3
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.69 = 23
Check. A different route: divide in small steps (÷ 3). 69 → 23. We reach the same fraction. As decimals, 69 = 0.6667 and 23 = 0.6667. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.1015
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 10: 1, 2, 5, 10
Factors of 15: 1, 3, 5, 15 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 5
HCF = 5 - Divide the top by 5. Whatever we do to the top we must also do to the bottom.10 ÷ 5 = 2
- Divide the bottom by 5. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.15 ÷ 5 = 3
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.1015 = 23
Check. A different route: divide in small steps (÷ 5). 1015 → 23. We reach the same fraction. As decimals, 1015 = 0.6667 and 23 = 0.6667. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.1216
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 4
HCF = 4 - Divide the top by 4. Whatever we do to the top we must also do to the bottom.12 ÷ 4 = 3
- Divide the bottom by 4. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.16 ÷ 4 = 4
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.1216 = 34
Check. A different route: divide in small steps (÷ 2, then ÷ 2). 1216 → 68 → 34. We reach the same fraction. As decimals, 1216 = 0.75 and 34 = 0.75. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.1824
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 3, 6
HCF = 6 - Divide the top by 6. Whatever we do to the top we must also do to the bottom.18 ÷ 6 = 3
- Divide the bottom by 6. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.24 ÷ 6 = 4
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.1824 = 34
Check. A different route: divide in small steps (÷ 2, then ÷ 3). 1824 → 912 → 34. We reach the same fraction. As decimals, 1824 = 0.75 and 34 = 0.75. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.2050
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 50: 1, 2, 5, 10, 25, 50 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 5, 10
HCF = 10 - Divide the top by 10. Whatever we do to the top we must also do to the bottom.20 ÷ 10 = 2
- Divide the bottom by 10. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.50 ÷ 10 = 5
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.2050 = 25
Check. A different route: divide in small steps (÷ 2, then ÷ 5). 2050 → 1025 → 25. We reach the same fraction. As decimals, 2050 = 0.4 and 25 = 0.4. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.2842
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 28: 1, 2, 4, 7, 14, 28
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 7, 14
HCF = 14 - Divide the top by 14. Whatever we do to the top we must also do to the bottom.28 ÷ 14 = 2
- Divide the bottom by 14. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.42 ÷ 14 = 3
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.2842 = 23
Check. A different route: divide in small steps (÷ 2, then ÷ 7). 2842 → 1421 → 23. We reach the same fraction. As decimals, 2842 = 0.6667 and 23 = 0.6667. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.3660
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 2, 3, 4, 6, 12
HCF = 12 - Divide the top by 12. Whatever we do to the top we must also do to the bottom.36 ÷ 12 = 3
- Divide the bottom by 12. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.60 ÷ 12 = 5
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.3660 = 35
Check. A different route: divide in small steps (÷ 2, then ÷ 2, then ÷ 3). 3660 → 1830 → 915 → 35. We reach the same fraction. As decimals, 3660 = 0.6 and 35 = 0.6. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.4575
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 75: 1, 3, 5, 15, 25, 75 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 3, 5, 15
HCF = 15 - Divide the top by 15. Whatever we do to the top we must also do to the bottom.45 ÷ 15 = 3
- Divide the bottom by 15. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.75 ÷ 15 = 5
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.4575 = 35
Check. A different route: divide in small steps (÷ 3, then ÷ 5). 4575 → 1525 → 35. We reach the same fraction. As decimals, 4575 = 0.6 and 35 = 0.6. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify by dividing the top and the bottom by the same number: the highest common factor (HCF).
- Say what we are trying to do. To simplify means to write the fraction with smaller numbers without changing its value. We do this by dividing the top and the bottom by a number that goes into both.6381
- List the factors of the top and of the bottom. A factor of a number divides into it exactly. We want factors that the two numbers share.Factors of 63: 1, 3, 7, 9, 21, 63
Factors of 81: 1, 3, 9, 27, 81 - Pick the highest common factor (HCF). The HCF is the biggest number in both lists. Dividing by it simplifies completely in one go.Common factors: 1, 3, 9
HCF = 9 - Divide the top by 9. Whatever we do to the top we must also do to the bottom.63 ÷ 9 = 7
- Divide the bottom by 9. Dividing top and bottom by the same number keeps the fraction the same size but with smaller numbers.81 ÷ 9 = 9
- Write the simplified fraction. Check that nothing more can be divided out: the HCF was used, so the top and bottom now share only the factor 1.6381 = 79
Check. A different route: divide in small steps (÷ 3, then ÷ 3). 6381 → 2127 → 79. We reach the same fraction. As decimals, 6381 = 0.7778 and 79 = 0.7778. ✓
Watch out. Divide the top and the bottom. If you divide only one of them you change the value of the fraction.
Method. Simplify each fraction. If it simplifies to the same fraction as the target, it is equivalent to it.
- Decide the test. To check whether a fraction is equal to 23, simplify it fully and compare with 23.Target: 23
- Simplify each fraction fully. Divide the top and bottom by their highest common factor.46 ÷ 2 = 23
610 ÷ 2 = 35
1015 ÷ 5 = 23
812 ÷ 4 = 23
1220 ÷ 4 = 35 - Compare each with the target. Keep those that simplify to 23.46: = 23 ✓
610: is 35, not 23 ✗
1015: = 23 ✓
812: = 23 ✓
1220: is 35, not 23 ✗ - Write the answer. List the fractions that are equal to the target.46, 1015, 812
Check. A different test, as decimals: 46 = 0.6667; 610 = 0.6; 1015 = 0.6667; 812 = 0.6667; 1220 = 0.6. Those equal to 0.6667 are the ones listed. ✓
Watch out. 610 is a close one: it simplifies to 35, not 23.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. There are 60 minutes in 1 hour, so compare 15 minutes with 60 minutes. Both are already in minutes.Part = 15, whole = 60
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1560
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 15: 1, 3, 5, 15
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
HCF = 15 - Divide the top and the bottom by 15. Dividing both by the same number keeps the value the same but gives smaller numbers.15 ÷ 15 = 1 and 60 ÷ 15 = 4
1560 = 14
Check. 15 ÷ 60 = 0.25 and 14 = 0.25. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. 1 m = 100 cm. Both amounts must be in the same units, so change the metre to centimetres.Part = 40, whole = 100
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).40100
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
HCF = 20 - Divide the top and the bottom by 20. Dividing both by the same number keeps the value the same but gives smaller numbers.40 ÷ 20 = 2 and 100 ÷ 20 = 5
40100 = 25
Check. 40 ÷ 100 = 0.4 and 25 = 0.4. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. 1 kg = 1000 g. Both amounts must be in the same units, so change the kilogram to grams.Part = 250, whole = 1000
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).2501000
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 250: 1, 2, 5, 10, 25, 50, 125, 250
Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
HCF = 250 - Divide the top and the bottom by 250. Dividing both by the same number keeps the value the same but gives smaller numbers.250 ÷ 250 = 1 and 1000 ÷ 250 = 4
2501000 = 14
Check. 250 ÷ 1000 = 0.25 and 14 = 0.25. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. 1 minute = 60 seconds. Both must be in seconds.Part = 45, whole = 60
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).4560
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
HCF = 15 - Divide the top and the bottom by 15. Dividing both by the same number keeps the value the same but gives smaller numbers.45 ÷ 15 = 3 and 60 ÷ 15 = 4
4560 = 34
Check. 45 ÷ 60 = 0.75 and 34 = 0.75. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. 1 year = 12 months. Both must be in months.Part = 8, whole = 12
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).812
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
HCF = 4 - Divide the top and the bottom by 4. Dividing both by the same number keeps the value the same but gives smaller numbers.8 ÷ 4 = 2 and 12 ÷ 4 = 3
812 = 23
Check. 8 ÷ 12 = 0.6667 and 23 = 0.6667. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. £2.40 = 240p. Both must be in pence.Part = 36, whole = 240
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).36240
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
HCF = 12 - Divide the top and the bottom by 12. Dividing both by the same number keeps the value the same but gives smaller numbers.36 ÷ 12 = 3 and 240 ÷ 12 = 20
36240 = 320
Check. 36 ÷ 240 = 0.15 and 320 = 0.15. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. A part as a fraction of the whole: write the part over the whole, then simplify.
- Make sure both amounts are in the same units. Both numbers are counts of pupils, so the units already match.Part = 18, whole = 30
- Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).1830
- List the factors of the top and the bottom. We want the highest common factor (HCF) so we can simplify.Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
HCF = 6 - Divide the top and the bottom by 6. Dividing both by the same number keeps the value the same but gives smaller numbers.18 ÷ 6 = 3 and 30 ÷ 6 = 5
1830 = 35
Check. 18 ÷ 30 = 0.6 and 35 = 0.6. They match. ✓
Watch out. Put the part on top and the whole on the bottom, and always use the same units for both.
Method. Equivalent fractions come from multiplying or dividing the top and bottom by the same number. Adding the same number does not keep the value.
- Test what Tom did. He added 3 to the top and 3 to the bottom.37 → 3 + 37 + 3 = 610
- Compare the values as decimals. If two fractions are equivalent they have the same value.37 = 0.43 (to 2 d.p.) but 610 = 0.6
- Compare by simplifying. 610 simplifies to 35. Is that the same as 37? The denominators 5 and 7 are different and the tops are the same, so no.610 = 35, which is not 37
- Show a correct equivalent fraction. Multiply the top and bottom by 2.37 × 2 = 614, so 37 = 614
Check. Cross-multiply Tom's claim: 3 × 10 = 30 but 7 × 6 = 42. These are not equal, so 37 ≠ 610. ✓
Watch out. Adding to the top and the bottom changes the fraction. Multiply or divide.
Method. Equivalent to 2/3 means the top is 2k and the bottom is 3k for some multiplier k.
- Write the fraction using a multiplier k. To make an equivalent fraction we multiply the top 2 and the bottom 3 by the same number k.2k3k
- Use the sum 35. The numerator and denominator add up to 35.2k + 3k = 35
5k = 35 - Solve for k. Divide both sides by 5.k = 35 ÷ 5 = 7
- Find the numerator and the denominator. Multiply 2 and 3 by the multiplier 7.2 × 7 = 14 and 3 × 7 = 21
1421
Check. 14 + 21 = 35 and 1421 simplifies to 23 (divide both by 7). ✓
Watch out. Use the same multiplier on the top and the bottom. The sum of 35 tells us the total number of parts, 2k + 3k.
Method. A fraction is in simplest form when its top and bottom have no common factor except 1.
- Look for a common factor. 17 is a prime number, so its only factors are 1 and 17. The only way 17 and 51 share a factor other than 1 is if 17 divides into 51.Test: 51 ÷ 17
- Do the division. If it divides exactly, 17 is a common factor.17 × 3 = 51, so 51 ÷ 17 = 3 exactly
- Divide the top and the bottom by 17. Dividing both by the same number does not change the value.17 ÷ 17 = 1 and 51 ÷ 17 = 3
1751 = 13 - Write the conclusion. It simplifies, so it was not in simplest form.No, 1751 is not in simplest form. Its simplest form is 13.
Check. As decimals, 1751 = 0.333… and 13 = 0.333…. They are equal. ✓
Watch out. Big numbers can still simplify. Test whether the smaller number divides into the larger one.
Method. Divide the top and the bottom by their highest common factor (HCF).
- List what we know about the numbers. 84 = 2 × 42 and 126 = 3 × 42, so 42 divides into both.Both are multiples of 42
- Find the HCF of 84 and 126. Check that no bigger number works. The multiples of 42 near these are 84 (42 × 2) and 126 (42 × 3), and 2 and 3 have no common factor except 1, so 42 is the highest.HCF = 42
- Divide the top and the bottom by 42. The same number is divided into both.84 ÷ 42 = 2 and 126 ÷ 42 = 3
- Write the answer. Show that you found the HCF first, as the question asks.84126 = 23
Check. A different route: divide in steps. 84126 → (÷ 2) 4263 → (÷ 3) 1421 → (÷ 7) 23. Same answer. ✓
Watch out. If you divide by a common factor that is not the HCF, you must keep simplifying until no common factor is left.