SHSAT: Fractions of a changing whole
Track which whole each fraction refers to, update it after every step, and work backward from a part to the whole.
When a problem takes a fraction, then another fraction, how do you know what each fraction is taken of?
Short answer
Each fraction is taken of a specific whole. Name that whole before you calculate, and update it whenever something is removed or added.
Know first: Multiplying a whole number by a fraction, Dividing by a fraction
Step 1 of 5
Every fraction is a fraction of something
A fraction on its own is incomplete. One third of 90 and one third of 60 are different amounts. Before you calculate, ask: one third of what?
Word problems often change the whole partway through. Something is removed, and the next fraction is taken of what's left. Or a tank's fractions are fractions of its capacity, not of the water inside.
| The question says | The whole is | Example |
|---|---|---|
| 1/3 of the remaining | what is left after the last step | 60 left, so 1/3 is 20 |
| the tank is 2/5 full | the tank's full capacity | capacity 50 L holds 20 L |
| 21 green are 3/7 of the bag | the unknown total | 21 ÷ 3/7 = 49 marbles |
| 1/3 of the boys wear glasses | the boys only | boys are 2/5 of the class, so 2/5 × 1/3 = 2/15 of the class |
When a part and its fraction are given, divide the part by the fraction to find the whole.
Step 2 of 5
Update the whole after each step
Worked example: Books on a cart
A cart holds 120 books. One sixth of them are shelved. Then 3/10 of the books still on the cart are shelved. How many books remain on the cart?
- A30
- B64
- C70
- D100
- 1
First whole: 120
1/6 of 120 is 20, so 120 − 20 = 100 books remain.
- 2
Second whole: 100
3/10 of 100 is 30. This fraction is taken of the 100 left, not of 120.
- 3
Finish
100 − 30 = 70 books remain.
- 4
See why the other choices are there
30 is the number shelved in the second step. 64 takes 3/10 of 120 instead of 100. 100 stops after the first step.
Answer
70 books
Wrong: A tank goes from 3/4 full to 1/3 full when 25 liters are used, so capacity = 25 ÷ 1/3 = 75 liters.
Right: The 25 liters are 3/4 − 1/3 = 5/12 of capacity, so capacity = 25 ÷ 5/12 = 60 liters.
The amount used matches the change in the fraction, not the fraction at the end. Check: 3/4 of 60 is 45, and 45 − 25 = 20, which is 1/3 of 60.
Step 3 of 5
Find the whole before you answer
Check yourself · Question 1
Of 45 club members, 2/5 play chess. Then 3 chess players leave the club. What fraction of the remaining members play chess?
Answer: B
Chess players: 2/5 of 45 = 18. After 3 leave, 15 chess players remain among 45 − 3 = 42 members. 15/42 = 5/14.
Trap answer: 1/3
Keeping the old denominator
- Why it looks right
- You correctly took 3 away from the chess players, so the work feels finished.
- Why it's wrong
- The 3 people who left were also club members. The whole is now 42, not 45.
- The right answer
- 5/14: 15 chess players out of 42 members.
Check yourself · Question 2
A crate is 5/6 full of apples. After 24 kilograms are sold, it is 1/2 full. What is the crate's full capacity in kilograms?
Answer: D
The 24 kilograms equal 5/6 − 1/2 = 5/6 − 3/6 = 2/6 = 1/3 of capacity. Capacity = 24 ÷ 1/3 = 72. Check: 5/6 of 72 is 60, and 60 − 24 = 36, which is half of 72.
Common mistake
I take every fraction of the number at the start of the problem.
- Why it's tempting
- The starting number is the only total written in the question, so it feels like the whole.
- Do this instead
- After each removal, write down what's left. The next fraction is taken of that new amount unless the question says otherwise.
Step 4 of 5
What to remember
Remember
- Ask "of what?" for every fraction, and write that whole next to it.
- After a removal, the next fraction uses what's left, and a group's fraction uses its new total.
- To find a whole, divide the known part by its fraction. For capacity, use the change in fraction.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what was the whole for the first fraction, and what was the whole for the second fraction?