Algebra Word Problems Worksheets - 15 Worksheets Library - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

1. Family Ages

Problem: The sum of the ages of a father, mother, and son is 75. The father is three times as old as the son, and the mother is five years older than the father. Find the ages.

* Step 1: Let the son's age be $s$.
* Step 2: The father is 3 times the son's age, so Father = $3s$.
* Step 3: The mother is 5 years older than the father, so Mother = $3s + 5$.
* Step 4: Add them all together to equal 75:
$$s + 3s + (3s + 5) = 75$$
* Step 5: Combine the $s$ terms:
$$7s + 5 = 75$$
* Step 6: Subtract 5 from both sides:
$$7s = 70$$
* Step 7: Divide by 7:
$$s = 10$$
* Step 8: Calculate the others:
* Son = 10
* Father = $3 \times 10 = 30$
* Mother = $30 + 5 = 35$
* Check: $10 + 30 + 35 = 75$. Correct.

2. Sibling Ages

Problem: Sarah is three years older than Tom. The sum of their ages is 23.

* Step 1: Let Tom's age be $t$.
* Step 2: Sarah is 3 years older, so Sarah = $t + 3$.
* Step 3: Add their ages:
$$t + (t + 3) = 23$$
* Step 4: Combine terms:
$$2t + 3 = 23$$
* Step 5: Subtract 3:
$$2t = 20$$
* Step 6: Divide by 2:
$$t = 10$$
* Step 7: Calculate Sarah's age:
$$10 + 3 = 13$$

3. Parent-Child Ratios

Problem: Ratio of parent to child is 5:1. Parent is 40.

* Step 1: The ratio means for every 5 years the parent has, the child has 1.
* Step 2: Since the parent is 40, we divide by 5 to find what "1 part" equals.
$$40 / 5 = 8$$
* Step 3: The child is 1 part, so the child is 8 years old.

4. Expenses

Problem: Rent is $\$1,200$, utilities cost $\$350$, and groceries cost three times utilities. Total monthly expense?

* Step 1: Calculate groceries:
$$3 \times \$350 = \$1,050$$
* Step 2: Add everything up:
$$\$1,200 (\text{Rent}) + \$350 (\text{Utilities}) + \$1,050 (\text{Groceries})$$
* Step 3: Sum:
$$1200 + 350 + 1050 = 2600$$

5. Inheritance Sharing

Problem: Estate is $\$300,000$. Father gets 3x son, Mother gets 2x son.

* Step 1: Let the son's share be $x$.
* Step 2: Father gets $3x$, Mother gets $2x$.
* Step 3: Total equation:
$$x + 3x + 2x = 300,000$$
* Step 4: Combine $x$'s:
$$6x = 300,000$$
* Step 5: Divide by 6:
$$x = 50,000$$
* Step 6: Calculate shares:
* Son: $\$50,000$
* Father: $3 \times 50,000 = \$150,000$
* Mother: $2 \times 50,000 = \$100,000$

6. Family Savings

Problem: Total savings $\$20,000$. Father saves 3x Mother. Daughter saves half of Mother.

* Step 1: Let Mother's savings be $m$.
* Step 2: Father = $3m$. Daughter = $0.5m$ (or $m/2$).
* Step 3: Equation:
$$m + 3m + 0.5m = 20,000$$
* Step 4: Combine terms ($1 + 3 + 0.5 = 4.5$):
$$4.5m = 20,000$$
* Step 5: Solve for $m$:
$$m = 20,000 / 4.5 = 4,444.44$$ (repeating)
* Step 6: Calculate others based on Mother ($\$4,444.44$):
* Father: $3 \times 4,444.44 = 13,333.33$
* Daughter: $4,444.44 / 2 = 2,222.22$
*(Note: Due to rounding decimals, the cents might vary slightly, but these are the correct values).*

7. Family Investments

Problem: Total $\$10,000$. Stocks are 3x Bonds.

* Step 1: Let Bonds be $b$.
* Step 2: Stocks are $3b$.
* Step 3: Equation:
$$b + 3b = 10,000$$
* Step 4: Combine:
$$4b = 10,000$$
* Step 5: Divide by 4:
$$b = 2,500$$
* Step 6: Calculate Stocks:
$$3 \times 2,500 = 7,500$$

8. Height

Problem: Sum of heights is 160 cm. Father is 20 cm taller than son.

* Step 1: This problem only gives two variables (Father and Son) but mentions "each family member" in the question. Assuming it refers only to the two people mentioned in the math setup:
* Step 2: Let Son's height be $s$.
* Step 3: Father's height is $s + 20$.
* Step 4: Equation:
$$s + (s + 20) = 160$$
* Step 5: Combine:
$$2s + 20 = 160$$
* Step 6: Subtract 20:
$$2s = 140$$
* Step 7: Divide by 2:
$$s = 70$$
* Step 8: Father's height:
$$70 + 20 = 90$$
*(Note: A 70cm son and 90cm father is physically impossible for humans, suggesting a typo in the original worksheet numbers, but mathematically this is the correct solution to the text provided).*

9. Family Vehicle Expenses

Problem: Total $\$300$. Father pays 2x Mother. Son pays half of Mother.

* Step 1: Let Mother's payment be $m$.
* Step 2: Father = $2m$. Son = $0.5m$.
* Step 3: Equation:
$$m + 2m + 0.5m = 300$$
* Step 4: Combine ($1 + 2 + 0.5 = 3.5$):
$$3.5m = 300$$
* Step 5: Solve for $m$:
$$m = 300 / 3.5 = 85.71$$ (rounded)
* Step 6: Calculate others:
* Mother: $\$85.71$
* Father: $2 \times 85.71 = \$171.42$
* Son: $85.71 / 2 = \$42.86$

10. Vacation Budget

Problem: Total budget $\$5,000$. Parents contribute twice as much as their child.

* Step 1: "Parents" usually implies two people, but "contribute twice as much as their child" treats the parents as a single unit or implies each parent contributes individually. Given standard word problem phrasing, it usually means: (Parent 1 + Parent 2) vs Child, OR Each Parent vs Child.
* *Interpretation A (Most likely for simple algebra):* The group "Parents" puts in money, and the "Child" puts in money. But usually, it lists individuals. Let's assume there are 2 parents and 1 child.
* Let Child's contribution = $c$.
* If "The parents" (plural) contribute twice as much as the child, does it mean Mom = $2c$ and Dad = $2c$? Or Mom+Dad = $2c$?
* Standard interpretation: Each parent contributes twice as much as the child.
* Child = $c$
* Mother = $2c$
* Father = $2c$
* Step 2: Equation:
$$c + 2c + 2c = 5,000$$
* Step 3: Combine:
$$5c = 5,000$$
* Step 4: Divide by 5:
$$c = 1,000$$
* Step 5: Calculate parents:
$$2 \times 1,000 = 2,000$$ each.

Final Answer:
1. Son: 10, Father: 30, Mother: 35
2. Tom: 10, Sarah: 13
3. Child: 8 years old
4. Total Expense: $2,600
5. Son: $50,000, Mother: $100,000, Father: $150,000
6. Mother: $4,444.44, Father: $13,333.33, Daughter: $2,222.22
7. Bonds: $2,500, Stocks: $7,500
8. Son: 70 cm, Father: 90 cm
9. Mother: $85.71, Father: $171.42, Son: $42.86
10. Child: $1,000, Each Parent: $2,000