Reasoning with Equations & Inequalities — Practice
Before you solve anything, translate. Most of the difficulty in a word problem is turning English into algebra.
What these questions cover
These questions use 2 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
Problem 1
Which expression means "5 more than n"?
- n + 5
- 5 - n
- n - 5
- 5n
Worked answer
n + 5
n + 5
Problem 2
Which expression means "7 less than x"?
- 7 - x
- x - 7
- x + 7
- x7
Worked answer
x - 7
x - 7
A very common slip is writing 7 - x — but you are starting from x.
Problem 3
Twice a number is 18. Find the number.
Worked answer
9
2n = 18
n = 9
Problem 4
Which symbol means "at least"?
- <
- >
- ≤
- ≥
Worked answer
≥
≥
Problem 5
Three times a number, plus 2, is 17. Find the number.
Worked answer
5
3n + 2 = 17
3n = 15
n = 5
Problem 6
A number decreased by 9 is 23. Find the number.
Worked answer
32
n - 9 = 23
n = 32
Problem 7
An 84 cm rope is cut into two pieces, one three times as long as the other. Find the shorter piece.
Worked answer
21
s + 3s = 84
4s = 84
s = 21, and the longer piece is 63 ✓
Problem 8
The sum of two consecutive integers is 47. Find the smaller one.
Worked answer
23
n + (n + 1) = 47
2n + 1 = 47
2n = 46, so n = 23 and the other is 24.
Problem 9
"5 less than twice a number is 19." Find the number.
Worked answer
12
2n - 5 = 19
2n = 24
n = 12
Problem 10
Devon has $50. A ticket costs $12 and snacks are $4 each. At most how many snacks can he buy?
Worked answer
9
12 + 4s ≤ 50
4s ≤ 38
s ≤ 9.5
He cannot buy half a snack, so 9.
Problem 11
A rectangle's length is 3 more than its width, and its perimeter is 26. Find the width.
Worked answer
5
2w + 2(w + 3) = 26
4w + 6 = 26
4w = 20, so w = 5 and the length is 8 ✓
Problem 12
The sum of three consecutive even integers is 84. Find the smallest.
Worked answer
26
n + (n+2) + (n+4) = 84
3n + 6 = 84
3n = 78, so n = 26. The three are 26, 28, 30 ✓
Problem 13
Ana is 4 years older than Ben. In 3 years, their ages will sum to 30. How old is Ben now?
Worked answer
10
In 3 years: Ben is b + 3, Ana is b + 7
(b + 3) + (b + 7) = 30
2b + 10 = 30
b = 10 — Ben is 10, Ana is 14 ✓
Problem 14
A jar holds 20 coins, only nickels and dimes, worth $1.55. How many dimes are there?
Worked answer
11
5n + 10(20 - n) = 155
5n + 200 - 10n = 155
-5n = -45, so n = 9 nickels
Dimes: 20 - 9 = 11. Check: 45 + 110 = 155 ✓
Problem 15
A number is tripled, then 7 is subtracted. The result equals the number increased by 13. Find the number.
Worked answer
10
Left: 3n - 7. Right: n + 13
3n - 7 = n + 13
2n = 20
n = 10. Check: 23 = 23 ✓
Problem 16
A school has 178 students going on a trip. Each bus holds 40. How many buses are needed?
Worked answer
5
178 ÷ 40 = 4.45
Four buses hold only 160 students
Round UP: 5 buses, because the extra 18 students still need a ride.