TACHS: Budget limits: the greatest whole number that fits
Write a budget as an inequality, solve it, and round correctly, including fees, extra people, change and the greatest price.
How do you find the most people or items a budget can pay for when there is a fixed fee?
Short answer
Write fee + price × number ≤ budget, subtract the fee, and divide. When you are counting people or items, round down, then check that number and the next one.
Know first: Solving a two-step equation, Dividing with a remainder
Step 1 of 6
A budget is an inequality
A group with at most $90 that pays a $12 fee plus $7 per student needs 12 + 7n ≤ 90. Solve it like an equation: subtract the fee, then divide by the price.
The result is often a decimal. You cannot bring part of a student, so round down to the greatest whole number that still fits. Rounding up always breaks the budget.
| You are finding | Finish by |
|---|---|
| A number of people or items | Rounding down to a whole number |
| A greatest price | Keeping dollars and cents; do not round up |
| Change from a payment | Payment − total cost, exactly |
| Students when adults also need tickets | Counting all tickets first, then subtracting the adults |
Read what the answer counts before you round.
Step 2 of 6
Subtract the fee, divide, round down
Worked example: A museum trip
A museum trip costs a $12 bus fee plus $7 for each student. The class can spend no more than $90. What is the greatest number of students who can go?
- A11
- B12
- C13
- D78
- 1
Write the inequality
12 + 7n ≤ 90.
- 2
Subtract the fee
7n ≤ 78.
- 3
Divide
n ≤ 78 ÷ 7, which is about 11.1.
- 4
Round down and check
11 students cost 12 + 77 = $89, which fits. 12 students cost 12 + 84 = $96, which is over.
- 5
See why the other choices are there
12 rounds 11.1 up and breaks the budget. 13 ignores the bus fee: 90 ÷ 7 is about 12.9, rounded up. 78 is the money left after the fee, not a number of students.
Answer
11 students can go.
Students and helpers
A club has $100. It pays a $10 room fee, and every person needs an $8 ticket. Three adult helpers are coming. What is the greatest number of students who can go?
- 1
Money for tickets: 100 − 10 = $90.
- 2
Tickets: 90 ÷ 8 = 11.25, so 11 tickets.
- 3
So 11 students can go.
What went wrong
The 3 adult helpers also need tickets, so not all 11 tickets go to students.
The fix
11 tickets in all, minus 3 for the helpers, leaves 8 students. Check: 11 tickets cost 88 + 10 = $98, and a 12th ticket would make it $106.
Step 3 of 6
When the unknown is a price
Some questions fix the number of tickets and ask for the greatest price. The same inequality works, but now the answer is money, so keep the cents.
A group has $60 for an $8 fee and 8 tickets: 8 + 8p ≤ 60, so 8p ≤ 52 and p ≤ 6.50. The greatest price is $6.50. Rounding up to $7 would break the budget, and rounding down to $6 is not the greatest.
Step 4 of 6
Round only when you are counting
Check yourself · Question 1
A party room costs $15 plus $9 per guest. The host can spend at most $110. What is the greatest number of guests?
Answer: A
15 + 9g ≤ 110, so 9g ≤ 95 and g ≤ 95 ÷ 9, about 10.6. Round down: 10 guests cost $105. 11 guests would cost $114.
Trap answer: 11
Rounding up
- Why it looks right
- 10.6 is closer to 11 than to 10, and rounding to the nearest whole number is a strong habit.
- Why it's wrong
- The budget is a ceiling. Any whole number above 10.6 costs more than $110.
- The right answer
- 10 guests, for $105.
Check yourself · Question 2
A team has $73 to pay an $11 entry fee and buy 8 jerseys of the same price. What is the greatest price per jersey that keeps the total at or below $73?
Answer: B
11 + 8p ≤ 73, so 8p ≤ 62 and p ≤ 62 ÷ 8 = 7.75. Check: 11 + 8 × 7.75 = $73.
Check yourself · Question 3
A workshop charges $7 plus $4 per student. A group brings 9 students and pays with $60. How much change does it get?
Answer: A
Total: 7 + 4 × 9 = 7 + 36 = $43. Change: 60 − 43 = $17.
Common mistake
"95 ÷ 9 is about 10.6, and I always round to the nearest whole number."
- Why it's tempting
- Rounding to the nearest number is the rule you use most often.
- Do this instead
- A budget is a limit, so round down for people and items, every time. Then check your number and the next one up.
Step 5 of 6
What to remember
Remember
- Write fee + price × number ≤ budget, subtract the fee, then divide.
- Round down when counting people or items; keep the cents when finding a price.
- Count everyone who needs a ticket, price free-ticket deals in whole groups, and check your answer and the next number up.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what decimal did you get before rounding, and which wrong choice comes from rounding it up?