Translating Word Problems into Math
30 practice problems that start with simple phrases and build up to multi-step algebra, perimeter, and area problems. Try each one on paper first, then tap “Show answer” to check your work. Keep the Translating Keywords notes (PDF) open while you work.
Keyword Cheat Sheet
- Add ( + )
- plus, more than, the sum of, the total of, increased by, added to
- Subtract ( − )
- minus, less than, the difference of, less, decreased by, subtracted from
- Multiply ( × )
- times, the product of, twice, double, multiplied by, of
- Divide ( ÷ )
- the quotient of, divided by, the ratio of
- Powers
- the square of, squared, the cube of, cubed
- Equals ( = )
- equals, is, is the same as, yields, amounts to
Watch the order: “four less than a number” is x − 4, not 4 − x. Use parentheses when a sum or difference is multiplied: “three times the sum of a number and five” is 3(x + 5).
Perimeter and Area Formulas
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle | P = 2l + 2w | A = l × w |
| Square | P = 4s | A = s² |
| Triangle | P = a + b + c | A = ½ × b × h |
| Circle | C = 2πr | A = πr² |
l = length, w = width, s = side, b = base, h = height, r = radius. Perimeter is the distance around; area is the space inside, measured in square units.
Level 1: Write the Expression
Translate each phrase into an algebraic expression. Let x be the unknown number.
1.Nine more than a number.
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- “More than” means add.
Answer: x + 9
2.Five less than a number.
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- “Less than” flips the order: start with the number, then subtract 5.
Answer: x − 5
3.The product of seven and a number.
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- “Product” means multiply.
Answer: 7x
4.The quotient of a number and four.
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- “Quotient” means divide, in the order given.
Answer: x ÷ 4 (or x/4)
5.Twelve decreased by a number.
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- Start with 12, then take away the number.
Answer: 12 − x
6.Twice a number, increased by three.
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- “Twice a number” is 2x.
- “Increased by three” adds 3.
Answer: 2x + 3
7.Three times the sum of a number and five.
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- “The sum of a number and five” is a group: (x + 5).
- Multiply the whole group by 3.
Answer: 3(x + 5)
8.The square of a number, minus six.
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- “The square of a number” is x².
- “Minus six” subtracts 6.
Answer: x² − 6
Level 2: Write and Solve the Equation
Translate each sentence into an equation, then solve for the unknown.
9.A number increased by eight is twenty-three. Find the number.
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- x + 8 = 23
- Subtract 8 from both sides.
Answer: x = 15
10.Six times a number equals fifty-four. Find the number.
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- 6x = 54
- Divide both sides by 6.
Answer: x = 9
11.Four less than a number is eleven. Find the number.
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- x − 4 = 11
- Add 4 to both sides.
Answer: x = 15
12.The quotient of a number and three is seven. Find the number.
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- x ÷ 3 = 7
- Multiply both sides by 3.
Answer: x = 21
13.Nine less a number amounts to two. Find the number.
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- 9 − x = 2
- Subtract 9 from both sides: −x = −7.
Answer: x = 7
14.Seven more than twice a number is twenty-five. Find the number.
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- 2x + 7 = 25
- Subtract 7: 2x = 18.
- Divide by 2.
Answer: x = 9
15.Three fourths of a number is eighteen. Find the number.
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- “Of” means multiply: (3/4)x = 18.
- Multiply both sides by 4/3.
Answer: x = 24
16.Maya has $50 saved and adds $15 each week. After how many weeks will she have $140?
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- Let w be the number of weeks: 50 + 15w = 140.
- Subtract 50: 15w = 90.
- Divide by 15.
Answer: w = 6 weeks
Level 3: Perimeter and Area
Use the formulas above. Draw a picture and label what you know before you start.
17.A rectangle is 8 cm long and 5 cm wide. Find its perimeter and area.
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- P = 2(8) + 2(5) = 16 + 10
- A = 8 × 5
Answer: P = 26 cm, A = 40 cm²
18.A square has sides that are 7 inches long. Find its perimeter and area.
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- P = 4(7)
- A = 7²
Answer: P = 28 in, A = 49 in²
19.A triangle has a base of 10 m and a height of 6 m. Find its area.
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- A = ½ × 10 × 6
Answer: A = 30 m²
20.A square has a perimeter of 36 feet. Find the length of one side and the area.
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- 4s = 36
- Divide by 4: s = 9.
- A = 9²
Answer: s = 9 ft, A = 81 ft²
21.A rectangle has an area of 72 square feet and a width of 6 feet. Find its length and perimeter.
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- 6l = 72, so l = 12.
- P = 2(12) + 2(6)
Answer: l = 12 ft, P = 36 ft
22.The length of a rectangle is 3 inches more than its width. The perimeter is 30 inches. Find the length and width.
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- Width = w, length = w + 3.
- 2(w + 3) + 2w = 30
- 4w + 6 = 30, so 4w = 24 and w = 6.
- Length = 6 + 3.
Answer: Width = 6 in, length = 9 in
23.A garden’s length is twice its width. The perimeter is 48 meters. Find the length, width, and area.
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- Width = w, length = 2w.
- 2(2w) + 2w = 48, so 6w = 48 and w = 8.
- Length = 2(8) = 16.
- A = 16 × 8
Answer: Width = 8 m, length = 16 m, A = 128 m²
24.The sides of a triangle are x, x + 2, and x + 4 centimeters. The perimeter is 24 cm. Find each side.
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- x + (x + 2) + (x + 4) = 24
- 3x + 6 = 24, so 3x = 18 and x = 6.
Answer: 6 cm, 8 cm, and 10 cm
Level 4: Challenge
Multi-step problems that combine translating, formulas, and solving equations.
25.Five times the difference of a number and two is the same as three times the number plus four. Find the number.
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- “The difference of a number and two” is a group: (x − 2).
- 5(x − 2) = 3x + 4
- 5x − 10 = 3x + 4
- 2x = 14
Answer: x = 7
26.The length of a rectangle is 4 less than three times its width. The perimeter is 56 cm. Find the length, width, and area.
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- Width = w, length = 3w − 4.
- 2(3w − 4) + 2w = 56
- 8w − 8 = 56, so 8w = 64 and w = 8.
- Length = 3(8) − 4 = 20.
- A = 20 × 8
Answer: Width = 8 cm, length = 20 cm, A = 160 cm²
27.A triangle has an area of 42 square inches and a base of 12 inches. Find its height.
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- ½ × 12 × h = 42
- 6h = 42
- Divide by 6.
Answer: h = 7 in
28.A square and an equilateral triangle (all three sides equal) have the same perimeter. The square’s sides are 6 inches. How long is each side of the triangle?
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- Square perimeter: 4(6) = 24.
- Triangle perimeter: 3s = 24.
- Divide by 3.
Answer: s = 8 in
29.A circular garden has a radius of 5 feet. How much fencing goes around it, and how much ground does it cover? Use π ≈ 3.14.
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- Fencing is the circumference: C = 2π(5) = 10π.
- Ground covered is the area: A = π(5²) = 25π.
Answer: C ≈ 31.4 ft, A ≈ 78.5 ft²
30.A rectangular room is 12 feet wide. Carpet costs $3 per square foot plus a $50 installation fee. The total bill is $590. How long is the room?
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- Let l be the length. Area = 12l.
- 3(12l) + 50 = 590
- 36l + 50 = 590, so 36l = 540.
- Divide by 36.
Answer: l = 15 ft