What a fraction is: simplify, compare, order

A fraction is equal parts of one whole

How to solve them

1. 1

   Numerator and denominator: parts of a whole

   fraction = numerator ÷ denominator

   The denominator (the bottom) counts the equal parts in one whole. The numerator (the top) counts the parts you take. To read a fraction from a story, ask two questions: what is the whole? and how many of its parts?

   Part of a group

   A squad has 9 soldiers. 4 are on guard duty. What fraction are on guard?

   The whole is the 9 soldiers: 4/9

   As a division

   What is 3/4 as a decimal?

   3 ÷ 4 = 0.75. If the bottom fits into 100, it is faster to make it 100: 3/4 = 75/100 = 0.75.

   More than one whole

   What is 7/4 as a decimal?

   7 ÷ 4 = 1.75, more than 1

   Shortcut

   Check which group they ask about

   If 4 of the 9 soldiers are on guard, the other 5 are not. The fraction not on guard is 5/9. Tests ask for both, so read which group the question wants.

   Next step: Equal fractions ↓
2. 2

   Equivalent fractions: same amount, different numbers

   Equivalent fractions name the same amount. You make one by multiplying the top and the bottom by the same number. In the picture, the same bar is cut into 3 parts and into 6 parts. 1 of 3 parts and 2 of 6 parts cover the same length: 1/3 = 2/6.

   Multiplying the top and the bottom by 4 is the same as multiplying by 4/4, which is 1, so the value stays. Adding the same number to the top and the bottom does change the value. That is a trap you will see in the choices.

   1. 2/3
   2. (2 × 4) / (3 × 4)
   3. 8/12

   Example

   2/3 = ?/12

   1. The bottom went from 3 to 12. 12 ÷ 3 = 4, so the bottom was multiplied by 4.
   2. Do the same to the top: 2 × 4 = 8.
   3. So 2/3 = 8/12.

   Answer 8/12

   Next step: Simplify ↓
3. 3

   Simplify: divide by the greatest common factor

   Simplifying goes the other way: divide the top and the bottom by the same number. A factor of a number divides it with nothing left over: the factors of 12 are 1, 2, 3, 4, 6 and 12. A common factor divides both numbers. The greatest common factor (GCF) is the biggest one.

   A fraction is in simplest form when its top and bottom share no factor except 1.

   Factors of 24 and 36

   Number	Factors
   24	1, 2, 3, 4, 6, 8, 12, 24
   36	1, 2, 3, 4, 6, 9, 12, 18, 36
   In both	1, 2, 3, 4, 6, 12

   1. 24/36
   2. (24 ÷ 12) / (36 ÷ 12)
   3. 2/3

   Example

   Simplify 24/36

   1. The biggest number in both lists is 12, so the GCF is 12.
   2. Divide both: 24 ÷ 12 = 2 and 36 ÷ 12 = 3.
   3. 2 and 3 share no factor but 1, so you are done: 2/3.

   Answer 2/3

   Shortcut

   Can’t see the GCF? Divide in small steps

   Halve while both numbers are even, then try 3, then 5: 24/36 → 12/18 → 6/9 → 2/3. Same answer, more steps. Just don’t stop before the end.

   Next step: Compare ↓
4. 4

   Compare two fractions: same bottom, or cross-multiply

   Same denominator: the parts are the same size, so the bigger top wins. 7/12 is more than 5/12.

   Same numerator: the smaller bottom wins, because its parts are bigger. 3/5 is more than 3/8: fifths are bigger slices than eighths.

   Different tops and bottoms: rewrite both over the same bottom, or use the shortcut, cross-multiplying: multiply each top by the other fraction’s bottom. The bigger product is on the side of the bigger fraction.

   1. 4/7 vs. 5/9
   2. (4 × 9)/(7 × 9) vs. (5 × 7)/(9 × 7)
   3. 36/63 vs. 35/63
   4. 4/7 is bigger

   Example

   The shortcut: cross-multiply 4/7 and 5/9

   1. Top left times bottom right: 4 × 9 = 36. Write 36 under 4/7.
   2. Top right times bottom left: 5 × 7 = 35. Write 35 under 5/9.
   3. 36 is bigger than 35, so 4/7 is bigger. They are the same tops you got over 63, without writing the 63.

   Answer 4/7 > 5/9

   Next step: Put in order ↓
5. 5

   Put several fractions in order: one common bottom

   With three or more fractions, give them all the same bottom. Use the least common denominator (LCD): the smallest number that every bottom divides into. Then the tops tell you the order.

   To find it, count up by the biggest bottom until every bottom divides it. For 10, 5, 4 and 20, the biggest is 20, and 10, 5 and 4 all divide 20. The LCD is 20.

   Each fraction over 20

   Fraction	Multiply top and bottom by	Over 20
   3/10	2	6/20
   2/5	4	8/20
   1/4	5	5/20
   7/20	1	7/20

   Example

   Order 3/10, 2/5, 1/4 and 7/20 from least to greatest

   1. Over 20, the tops are 6, 8, 5 and 7.
   2. Put the tops in order: 5, 6, 7, 8.
   3. Go back to the original fractions: 1/4, 3/10, 7/20, 2/5.

   Answer 1/4 < 3/10 < 7/20 < 2/5

   Shortcut

   Compare with 1/2 first

   A fraction is more than 1/2 when its top is more than half its bottom. 4/9 is less than 1/2 (half of 9 is 4.5) and 5/8 is more (half of 8 is 4). Any fraction under 1/2 is smaller than any fraction over it.

   Next step: Mixed numbers ↓
6. 6

   Improper fractions and mixed numbers

   An improper fraction has a top as big as its bottom or bigger, like 17/5: it is one whole or more. A mixed number writes the same amount as wholes plus a fraction: 3 2/5, read “three and two fifths.”

   Improper to mixed: divide. The answer is the number of wholes, and the remainder goes on top of the same bottom. Mixed to improper: multiply the whole number by the bottom, then add the top.

   Example

   Write 17/5 as a mixed number

   1. How many 5s fit in 17? Three, because 3 × 5 = 15.
   2. The remainder is 17 − 15 = 2.
   3. Keep the same bottom, 5: 3 2/5.

   Answer 3 2/5

   1. 4 3/8
   2. (4 × 8)/8 + 3/8
   3. 32/8 + 3/8
   4. 35/8

   Next step: Three examples ↓
7. 7

   Three worked examples, from easy to test level

   Each one starts with the first try many people make, crossed out, and then the way that works. Read the wrong try too: it is the kind of answer the test puts in the choices.

   The test-level one mixes an improper fraction with mixed numbers. Rewrite everything in one form before you compare anything.

   Example · Easy

   Simplify 18/30.

   First try18/30 = 9/15, done

   1. Halving gives 9/15, but 9 and 15 still share the factor 3. 9/15 is equal to 18/30, just not in simplest form.
   2. Find the GCF. The factors of 18 are 1, 2, 3, 6, 9, 18. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. The biggest in both lists is 6.
   3. Divide both by 6: 18 ÷ 6 = 3 and 30 ÷ 6 = 5.
   4. 3 and 5 share no factor but 1: 3/5.

   Answer 3/5

   Example · Medium

   Which is greater, 7/8 or 9/11?

   First try9/11, because 9 and 11 are bigger than 7 and 8

   1. Bigger numbers do not mean a bigger fraction: elevenths are smaller pieces than eighths.
   2. Cross-multiply. 7 × 11 = 77 goes with 7/8.
   3. 9 × 8 = 72 goes with 9/11.
   4. 77 is bigger than 72, so 7/8 is greater. Check: over 88 they are 77/88 and 72/88.

   Answer 7/8

   Example · Test level

   A cook has three bags of rice. They weigh 13/5 lb, 2 1/2 lb and 2 7/10 lb. Which bag is the heaviest?

   First try13/5 lb, because 13 is the biggest number

   1. Put everything in one form. 5 fits into 13 two times (2 × 5 = 10), with 3 left over, so 13/5 = 2 3/5.
   2. All three bags have 2 whole pounds. Compare only the fractions: 3/5, 1/2 and 7/10.
   3. The LCD of 5, 2 and 10 is 10: 3/5 = 6/10, 1/2 = 5/10, and 7/10 stays 7/10.
   4. From light to heavy: 2 1/2, then 13/5, then 2 7/10. The heaviest is the 2 7/10 lb bag.

   Answer 2 7/10 lb

4 traps in the answer choices

The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.

15 questions in three levels

Warm-up first, then core, then test level. Pick an answer, then read why.

12 flashcards

Tap a card to flip it. Keep the ones you miss until they stick.