Solving problems with fractions (age 7+)
Solve problems involving counting in tenths, fractions of quantities, equivalence, fraction addition/subtraction, and fraction comparison
What a learner can do afterwards
- Solve a word problem requiring finding a fraction of a quantity
- Solve a problem that requires comparing or ordering fractions
- Choose and apply appropriate fraction knowledge to a multi-step problem
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You already know how to find a fraction of a number, add fractions, and tell which fraction is bigger. Real problems often mix these skills together, so you need to pick the right tool for each step.
Priya ate 3/10 of a cake. Tom ate 2/5 of the same size cake. To compare, change 2/5 into tenths: 2/5 is the same as 4/10. Since 4/10 is more than 3/10, Tom ate more cake.
When two fractions have different bottom numbers, change them so the bottom numbers match. Then it is easy to compare or add them.
Break a fraction problem into steps: work out the fraction of the amount first, match up the bottom numbers if you need to, then compare or combine.
More to read
4.4 Add and Subtract Fractions with Common Denominators
Adding fractions is easy when the bottoms are the same. Think of a pizza cut into 8 slices. If you eat 3 slices and your friend eats 2 slices, you ate 3/8 and your friend ate 2/8. Together you ate 5 slices, so 3/8 plus 2/8 is 5/8. The bottom stays 8 because the slices are the same size. Only the tops change.
Subtracting works the same way. If the pizza had 7/8 left and you eat 4/8, then 7 minus 4 is 3, so 7/8 minus 4/8 is 3/8.
Try it yourself: 2/5 plus 1/5. The bottom stays 5. Add the tops: 2 plus 1 is 3. The answer is 3/5. Now try 6/7 minus 2/7. The bottom stays 7. Take the tops away: 6 minus 2 is 4. The answer is 4/7.
Use equivalent fractions to compare
How do you know which fraction is bigger when the bottoms are different? You make the bottoms match, then compare the tops.
Example: which is bigger, 2/3 or 3/4?
Step 1: find a number both 3 and 4 fit into. 12 works.
Step 2: change 2/3 into twelfths. Multiply the top and bottom by 4: 2/3 is 8/12.
Step 3: change 3/4 into twelfth. Multiply the top and bottom by 3: 3/4 is 9/12.
Step 4: compare. 9/12 is more than 8/12, so 3/4 is bigger.
Fractions that look different but name the same amount are called equivalent fractions. 1/2 and 2/4 are equivalent. They are the same size. Making equivalent fractions is the key trick for comparing and for adding.
5.2: Addition and Subtraction of Fractions with Unlike Denominators
Sometimes the bottoms do not match. You cannot add 1/3 and 1/4 until you fix that. Here is how.
Step 1: find the smallest number both bottoms fit into. For 3 and 4, that number is 12.
Step 2: rename each fraction so its bottom is 12. 1/3 becomes 4/12. 1/4 becomes 3/12.
Step 3: now the bottoms match, so add the tops. 4 plus 3 is 7. The answer is 7/12.
Subtraction works the same way. For 5/6 minus 1/4, rename both as twelfths: 5/6 is 10/12 and 1/4 is 3/12. Then take away: 10 minus 3 is 7, so the answer is 7/12.
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