HSPT: Ratios: name the whole before you divide
Learn to tell a fraction of a group from a part-to-part ratio, so you always choose the right bottom number.
How do you know which number goes on the bottom when a question gives two groups?
Short answer
If the question asks for a fraction of the whole group, add the parts and put that total on the bottom. If it compares one part with another, the other part goes on the bottom.
Know first: Simplifying fractions, Adding whole numbers
Step 1 of 5
The question decides the bottom number
A ratio compares one part of a group with another part. A fraction of the group compares one part with the whole group. The same two counts can make either one, and HSPT answer choices usually include both.
So before you divide, ask what the bottom number should be. If the question says of all, of the total or of the class, the bottom number is everyone. If it says for every or compared with, the bottom number is the other part.
| Wording | Type | Bottom number |
|---|---|---|
| "What fraction of the team..." | Part to whole | Everyone on the team |
| "What fraction of all the coins..." | Part to whole | All the coins |
| "The ratio of cats to dogs..." | Part to part | The second part named |
| "For every 2 red, there are..." | Part to part | The other part |
When the question asks for a fraction of the group, add the parts first.
Step 2 of 5
Add the parts to get the whole
Worked example: A choir with two sections
A choir has 15 sopranos and 20 altos. What fraction of the choir are altos?
- A4/7
- B3/4
- C4/3
- D3/7
- 1
Name the whole
"Of the choir" means the bottom number is everyone: 15 + 20 = 35 singers.
- 2
Put the asked-for part on top
Altos: 20/35.
- 3
Simplify
Divide the top and bottom by 5: 20/35 = 4/7.
- 4
See why the other choices are there
3/4 is sopranos compared with altos, a part-to-part ratio. 4/3 is altos compared with sopranos, and a fraction of a group can never be more than 1. 3/7 is the fraction of the choir who are sopranos, the wrong group.
Answer
4/7 of the choir are altos.
Wrong: Altos ÷ sopranos = 20/15 = 4/3 of the choir
Right: Altos ÷ all singers = 20/35 = 4/7 of the choir
A fraction of the choir needs the whole choir on the bottom. Dividing by the other part gives a ratio, which answers a different question.
From a ratio to a fraction of the whole
The ratio of cats to dogs at a shelter is 5:3.
Parts in all: 5 + 3 = 8
Dogs are 3 of those 8 parts.
Fraction of the animals that are dogs: 3/8
Step 3 of 5
Pick the whole, then divide
Check yourself · Question 1
A box holds only pencils and pens: 9 pencils and 27 pens. What fraction of the items in the box are pencils?
Answer: B
The whole is 9 + 27 = 36 items. Pencils are 9/36, which simplifies to 1/4.
Trap answer: 1/3
Part over part
- Why it looks right
- Both numbers in the question are right there, so dividing one by the other feels finished.
- Why it's wrong
- The question says of the items in the box, so the bottom number must be all 36 items, not the 27 pens.
- The right answer
- 1/4, because 9 of the 36 items are pencils.
Check yourself · Question 2
The ratio of boys to girls in a club is 4:5. What fraction of the club are girls?
Answer: C
Think in parts: 4 + 5 = 9 parts in all, and girls are 5 of them. So girls make up 5/9 of the club.
Common mistake
"A ratio of 4:5 means the girls are 5/4 or 4/5 of the club."
- Why it's tempting
- The two numbers in the ratio look like a ready-made fraction.
- Do this instead
- A ratio only names parts. Add the parts to get the whole, then put the part you want over that total.
Step 4 of 5
What to remember
Remember
- Decide from the wording whether you need part to part or part to whole.
- For a fraction of a group, add every part to get the bottom number.
- Simplify at the end, and check that a fraction of a group is less than 1.
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 group, and which wrong choice comes from dividing one part by the other?