HSPT: Rates: find the cost of one
Learn to solve same-price and same-rate questions by finding the unit rate first, then scaling up.
If some items cost a certain amount, how do you find the cost of a different number of them?
Short answer
Divide the total cost by the number of items to get the cost of one. Then multiply by the number you need.
Know first: Dividing with money, Multiplying decimals
Step 1 of 5
Find the cost of one, then scale up
When items cost the same each, the total grows in step with the count. Twice as many items cost twice as much. This is a constant rate.
The surest method is to find the unit rate, the cost of one item, by dividing. Then multiply by the number you need. If the new count is a simple multiple of the old one, you can scale the total directly.
| Route | When to use it | Example: 2 lb cost $7; find 8 lb |
|---|---|---|
| Unit rate | Always works | $7 ÷ 2 = $3.50 per pound, times 8 = $28 |
| Scale factor | When the new count is a multiple of the old | 8 is 4 × 2, so 4 × $7 = $28 |
Both routes give the same answer. Pick the one with easier numbers.
Step 2 of 5
Divide for one, multiply for many
Worked example: Pens at a steady price
6 pens cost $9. At the same price, what do 10 pens cost?
- A$15
- B$13
- C$90
- D$5.40
- 1
Find the unit rate
$9 ÷ 6 = $1.50 per pen.
- 2
Multiply
10 × $1.50 = $15.
- 3
Sense check
10 pens is more than 6, so the cost should be more than $9. $15 fits.
- 4
See why the other choices are there
$13 adds $4 because 10 is 4 more than 6, which treats each extra pen as $1. $90 multiplies the $9 total by 10. $5.40 divides the wrong way, 9 × 6 ÷ 10.
Answer
$15
Find the wrong step
A student solves: 5 tickets cost $35. What do 8 tickets cost?
- 1
Finds the price of one ticket: $35 ÷ 5 = $7.
- 2
Notices that 8 is 3 more than 5 and adds $3: $35 + $3 = $38.
What went wrong
Each extra ticket costs $7, not $1. Adding the difference in count mixes tickets with dollars.
- 3
Answers $38.
The fix
Multiply the unit rate by the new count: 8 × $7 = $56.
Step 3 of 5
Use the unit rate
Check yourself · Question 1
3 bags of rice cost $8.40. At the same price, what do 5 bags cost?
Answer: B
One bag costs $8.40 ÷ 3 = $2.80. Five bags cost 5 × $2.80 = $14.00.
Trap answer: $10.40
Adding the extra count
- Why it looks right
- Two more bags, two more dollars, feels like a natural shortcut.
- Why it's wrong
- Each bag costs $2.80, so two more bags add $5.60, not $2.
- The right answer
- $14.00, because 5 × $2.80 = $14.00.
Check yourself · Question 2
A printer prints 45 pages in 3 minutes. At the same rate, how many minutes does it take to print 120 pages?
Answer: C
The rate is 45 ÷ 3 = 15 pages per minute. Then 120 ÷ 15 = 8 minutes.
Common mistake
"The new amount is 3 more, so I add 3 to the price."
- Why it's tempting
- Adding the difference is quick and the numbers are right there.
- Do this instead
- Prices grow by multiplying, not adding. Find the cost of one, then multiply by the new count.
Step 4 of 5
What to remember
Remember
- Divide the total by the count to find the unit rate.
- Multiply the unit rate by the new count; or scale the total when the counts are simple multiples.
- Check the size: more items should cost more, and in the same proportion.
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
What was the unit rate in the practice question, and what wrong answer comes from adding the difference in count?