Equation Word Problems

Show your work for each problem.

1.
Which equation represents: "A number increased by 7 equals 15"?
1. a) $x = 7 + 15$
2. b) $7x = 15$
3. c) $x - 7 = 15$
4. d) $x + 7 = 15$
Answer: $x + 7 = 15$
"A number increased by 7" translates to $x + 7$ . "Equals 15" means $= 15$ . Together: $x + 7 = 15$ .
2.
"5 less than a number" is written as:
1. a) $x + 5$
2. b) $5 - x$
3. c) $x - 5$
4. d) $5x$
Answer: $x - 5$
"5 less than a number" means we start with the number and subtract 5. So $x - 5$ . Remember: the number AFTER "than" comes first!
3.
A number plus 8 equals 20. What is the number?
Answer: 12
Equation: $x + 8 = 20$ . Subtract 8 from both sides: $x = 20 - 8 = 12$ .
4.
Three times a number is 27. What is the number?
Answer: 9
"Three times a number" is $3x$ . So $3x = 27$ . Divide by 3: $x = 9$ .
5.
Which phrase means $2x + 5$ ?
1. a)Twice a number minus 5
2. b)5 more than twice a number
3. c)5 times a number plus 2
4. d)Twice the sum of a number and 5
Answer: 5 more than twice a number
$2x + 5$ means: first double the number ( $2x$ ), then add 5. That's "5 more than twice a number".
6.
Tom is 4 years older than his sister. Together, their ages add up to 24. How old is Tom?
Answer: 14
1. Let x = sister's age. Write Tom's age in terms of x: x + 4
2. Write the equation for their ages adding to 24: x + (x + 4) = 24
3. Combine like terms: $x + x + 4 = \square x + 4$ . What number goes in the box? 2
4. Solve $2x + 4 = 24$ . What is x (sister's age)? 10
5. Tom's age is $x + 4$ . What is Tom's age? 14
7.
A number decreased by 9 equals 14. What is the number?
Answer: 23
$x - 9 = 14$ . Add 9 to both sides: $x = 14 + 9 = 23$ . Check: $23 - 9 = 14$ ✓
8.
A book costs 8 euros less than a game. Together they cost 42 euros. Which equation models this?
1. a) $x - (x - 8) = 42$
2. b) $x + (x - 8) = 42$
3. c) $x + (x + 8) = 42$
4. d) $8x = 42$
Answer: $x + (x - 8) = 42$
Let x = game price. Book = x - 8 (8 less). Together: game + book = $x + (x - 8) = 42$ .
9.
A rectangle's length is 5 cm more than its width. The perimeter is 46 cm. Find the width.
Answer: 9
1. Let w = width. Express the length in terms of w: w + 5
2. Perimeter formula: $2 \times \text{length} + 2 \times \text{width}$ . Write the equation: 2(w + 5) + 2w = 46
3. Expand and combine like terms: the equation becomes $\square w + 10 = 46$ . What number goes in the box? 4
4. Solve for w: 9
10.
Movie tickets cost 12 euros for adults and 7 euros for children. A group bought 8 tickets and paid 76 euros total. How many adult tickets were purchased?
Answer: 4
1. Let a = adult tickets. Express child tickets in terms of a: 8 - a
2. Write the cost equation (adult cost + child cost = 76): 12a + 7(8 - a) = 76
3. Expand $7(8-a)$ : 56 - 7a
4. Combine $12a - 7a$ : the equation $12a + 56 - 7a = 76$ becomes $\square a + 56 = 76$ . What number goes in the box? 5
5. Solve for a: 4
11.
Twice a number, decreased by 5, is 13. What is the equation?
1. a) $2x + 5 = 13$
2. b) $2(x - 5) = 13$
3. c) $x - 5 = 13$
4. d) $2x - 5 = 13$
Answer: $2x - 5 = 13$
"Twice a number" = $2x$ . "Decreased by 5" = subtract 5. So $2x - 5 = 13$ . Note: $2(x-5)$ would be "twice (a number decreased by 5)" - different meaning!
12.
The sum of three consecutive integers is 72. What is the smallest integer?
Answer: 23
Let integers be x, x+1, x+2. Sum: $x + (x+1) + (x+2) = 72$ . Simplify: $3x + 3 = 72$ , $3x = 69$ , $x = 23$ .
13.
A train travels at 80 km/h. A car starts 2 hours later traveling at 120 km/h on the same route. After how many hours will the car catch up to the train?
Answer: 4
1. Let t = car's travel time in hours. How long has the train been traveling? t + 2
2. Write the train's distance (speed × time): 80(t + 2)
3. Write the car's distance: 120t
4. When they meet, distances are equal. Write the equation: 120t = 80(t + 2)
5. Solve: $120t = 80t + 160$ . What is t? 4
14.
A piggy bank contains coins worth 4.40 euros. There are 8 more 20-cent coins than 50-cent coins. How many 50-cent coins are there?
Answer: 4
1. Let x = number of 50-cent coins. Express 20-cent coins in terms of x: x + 8
2. Value of 50-cent coins in euros: 0.50x
3. Value of 20-cent coins in euros: 0.20(x + 8)
4. Total value equation: 0.50x + 0.20(x + 8) = 4.40
5. Simplify and solve for x: 4
15.
Ana has 3 times as many stickers as Ben. Together they have 48 stickers. How many does Ana have?
1. a)36
2. b)24
3. c)12
4. d)16
Answer: 36
Let x = Ben's stickers. Ana = 3x. Together: $x + 3x = 48$ , so $4x = 48$ , $x = 12$ . Ana has $3(12) = 36$ stickers.
16.
A phone plan costs 15 euros per month plus 0.05 euros per text message. If the monthly bill is 23 euros, how many text messages were sent?
Answer: 160
1. Let m = number of messages. Write the equation for total cost: 15 + 0.05m = 23
2. Subtract 15 from both sides: 0.05m = 8
3. Divide by 0.05: 160
17.
Half of a number is 9. What is the number?
Answer: 18
$\frac{x}{2} = 9$ . Multiply both sides by 2: $x = 18$ .
18.
The product of a number and 4, increased by 6, equals 30. What is the number?
1. a)6
2. b)8
3. c)9
4. d)5
Answer: 6
$4x + 6 = 30$ . Subtract 6: $4x = 24$ . Divide by 4: $x = 6$ .