SHSAT: Ratios: count the parts, then find one part
Split a total by a ratio, find shares from a difference, link two ratios, and scale lengths and areas.
When a question gives a ratio and one extra fact, how do you find the actual amounts?
Short answer
Treat the ratio as equal-sized parts. Use the extra fact to find the size of one part, then multiply by the number of parts you need.
Know first: Multiplying and dividing whole numbers, Least common multiple
Step 1 of 5
A ratio is a count of equal parts
A ratio such as 3:7 doesn't tell you the amounts. It tells you the amounts are 3 parts and 7 parts of the same size. Call one part k: the amounts are 3k and 7k.
Every ratio question gives you one more fact, such as the total, the difference or one share. Use that fact to find k, then build whatever the question asks for.
| You're given | One part equals | Ratio 3:7 |
|---|---|---|
| the total | total ÷ sum of parts | total 150: 150 ÷ 10 = 15 |
| the difference | difference ÷ difference of parts | difference 60: 60 ÷ 4 = 15 |
| one share | that share ÷ its parts | smaller share 45: 45 ÷ 3 = 15 |
Once you know one part, any share is just a multiple of it.
Step 2 of 5
Count the parts before you divide
Worked example: Sharing stickers
Two friends share 150 stickers in the ratio 3:7. How many stickers are in the larger share?
- A45
- B50
- C75
- D105
- 1
Count the parts
3 + 7 = 10 parts.
- 2
Find one part
150 ÷ 10 = 15 stickers per part.
- 3
Build the share
The larger share is 7 parts: 7 × 15 = 105. Check: 45 + 105 = 150.
- 4
See why the other choices are there
45 is the smaller share. 50 divides by 3, one number of the ratio, instead of 10 parts. 75 splits 150 in half, which ignores the ratio.
Answer
105 stickers
Find the wrong step
Sand and gravel are mixed in the ratio 2:5. There are 36 more kilograms of gravel than sand. How many kilograms are there in all?
- 1
The difference is 5 − 2 = 3 parts.
- 2
One part is 36 ÷ 3 = 12 kilograms.
- 3
The total is 5 × 12 = 60 kilograms.
What went wrong
Five parts is only the gravel. The total includes the sand too.
The fix
The total is 2 + 5 = 7 parts, so 7 × 12 = 84 kilograms: 24 of sand and 60 of gravel.
Step 3 of 5
Use the extra fact to find one part
Check yourself · Question 1
Juice and water are mixed in the ratio 4:9. There are 35 more cups of water than juice. How many cups of juice are there?
Answer: B
The difference is 9 − 4 = 5 parts, so one part is 35 ÷ 5 = 7 cups. Juice is 4 parts: 4 × 7 = 28 cups. Check: water is 63, and 63 − 28 = 35.
Trap answer: 91
Answering a different question
- Why it looks right
- Finding the total is the last step in many ratio problems, so it feels like the finish line.
- Why it's wrong
- The question asks for juice only. 91 is all 13 parts.
- The right answer
- 28, the 4 parts of juice.
Check yourself · Question 2
The ratio of red pens to blue pens is 3:4. The ratio of blue pens to black pens is 6:5. There are 30 black pens. How many red pens are there?
Answer: B
Blue to black is 6:5, so blue = 30 × 6/5 = 36. Red to blue is 3:4, so red = 36 × 3/4 = 27.
Check yourself · Question 3
Two similar rectangles have side lengths in the ratio 3:4. The smaller one has area 45 square inches. What is the area of the larger one?
Answer: B
Areas scale by the square of the side ratio: (4/3)² = 16/9. 45 × 16/9 = 80 square inches.
Common mistake
I split the total in half, then adjust a little for the bigger share.
- Why it's tempting
- It's quick, and with close ratios such as 7:8 the halves look nearly right.
- Do this instead
- Halving is correct only for a 1:1 ratio. Add the parts, divide the total by that sum, and multiply by the share's part count.
Step 4 of 5
What to remember
Remember
- Write a ratio a:b as ak and bk, then use the given total, difference or share to find k.
- When two ratios share an item, rescale them so that item has the same number of parts.
- Similar figures: lengths scale by the ratio, areas by its square.
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, how many parts were there in all, and how big was one part?