Systems of Equations Word Problems Worksheets - Math Monks - Free Printable

Let's solve each of these systems of equations word problems step by step.

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Problem 1:

> A jar of coins has one third as many dimes as quarters. If the total value of coins is $5.10, how many dimes and quarters are in the jar?

#### Step 1: Define variables
- Let $ q $ = number of quarters
- Then, number of dimes = $ \frac{1}{3}q $

#### Step 2: Write the value equation
- Value of quarters: $ 0.25q $
- Value of dimes: $ 0.10 \times \frac{1}{3}q = \frac{0.10}{3}q $
- Total value: $ 0.25q + \frac{0.10}{3}q = 5.10 $

#### Step 3: Solve the equation
$$
0.25q + \frac{0.10}{3}q = 5.10
$$

Convert to decimals:
$$
0.25q + 0.0333\ldots q = 5.10
\Rightarrow (0.25 + 0.0333)q = 5.10
\Rightarrow 0.2833q = 5.10
$$

But better to use fractions:
- $ 0.25 = \frac{1}{4} $
- $ 0.10 = \frac{1}{10} $

So:
$$
\frac{1}{4}q + \frac{1}{10} \cdot \frac{1}{3}q = 5.10
\Rightarrow \frac{1}{4}q + \frac{1}{30}q = 5.10
$$

Find common denominator (LCM of 4 and 30 is 60):
$$
\frac{15}{60}q + \frac{2}{60}q = \frac{17}{60}q = 5.10
$$

Now solve:
$$
\frac{17}{60}q = 5.10
\Rightarrow q = 5.10 \times \frac{60}{17}
= \frac{510}{10} \times \frac{60}{17}
= 51 \times \frac{60}{17}
= \frac{3060}{17} = 180
$$

So, $ q = 18 $ → Wait! Let's recalculate carefully:

Wait: $ 5.10 = \frac{510}{100} = \frac{51}{10} $

So:
$$
\frac{17}{60}q = \frac{51}{10}
\Rightarrow q = \frac{51}{10} \times \frac{60}{17}
= \frac{51 \times 60}{10 \times 17}
= \frac{3060}{170}
= 18
$$

Yes! $ q = 18 $

Then dimes = $ \frac{1}{3} \times 18 = 6 $

#### ✔ Check:
- Quarters: 18 × $0.25 = $4.50
- Dimes: 6 × $0.10 = $0.60
- Total: $4.50 + $0.60 = $5.10 ✔️

Answer: 6 dimes and 18 quarters

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Problem 2:

> Ava has 14 coins that have a total value of $2.30. The coins are nickels and quarters. How many of each coin does he have?

#### Step 1: Define variables
- Let $ n $ = number of nickels
- Let $ q $ = number of quarters

#### Step 2: Set up equations
1. Total coins: $ n + q = 14 $
2. Total value: $ 0.05n + 0.25q = 2.30 $

Multiply second equation by 100 to eliminate decimals:
$$
5n + 25q = 230
$$

Divide by 5:
$$
n + 5q = 46 \quad \text{(Equation A)}
$$

From first equation: $ n = 14 - q $

Substitute into Equation A:
$$
(14 - q) + 5q = 46
\Rightarrow 14 + 4q = 46
\Rightarrow 4q = 32
\Rightarrow q = 8
$$

Then $ n = 14 - 8 = 6 $

#### ✔ Check:
- Nickels: 6 × $0.05 = $0.30
- Quarters: 8 × $0.25 = $2.00
- Total: $0.30 + $2.00 = $2.30 ✔️

Answer: 6 nickels and 8 quarters

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Problem 3:

> In an election, the winner received 25% more votes than the loser. If 5175 votes were cast, how many votes did each candidate get?

#### Step 1: Define variables
- Let $ x $ = number of votes the loser got
- Winner got 25% more: $ x + 0.25x = 1.25x $

Total votes: $ x + 1.25x = 2.25x = 5175 $

#### Step 2: Solve
$$
2.25x = 5175
\Rightarrow x = \frac{5175}{2.25}
$$

Convert 2.25 to fraction: $ \frac{9}{4} $

So:
$$
x = \frac{5175}{9/4} = 5175 \times \frac{4}{9} = \frac{20700}{9} = 2300
$$

Loser: 2300 votes
Winner: $ 1.25 \times 2300 = 2875 $

#### ✔ Check:
- 2300 + 2875 = 5175 ✔️
- Is 2875 = 2300 + 25% of 2300?
- 25% of 2300 = 575
- 2300 + 575 = 2875 ✔️

Answer: Loser got 2300 votes, winner got 2875 votes

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Problem 4:

> 2 times a number added to 10 times a second number is 20. 30 times the second number added to 3 times the first number is 45. What are the two numbers?

#### Step 1: Define variables
- Let $ x $ = first number
- Let $ y $ = second number

#### Step 2: Translate into equations
1. $ 2x + 10y = 20 $
2. $ 30y + 3x = 45 $

Simplify both equations:

Equation 1: Divide by 2 → $ x + 5y = 10 $ → $ x = 10 - 5y $

Equation 2: Divide by 3 → $ 10y + x = 15 $

Now substitute $ x = 10 - 5y $ into second simplified equation:
$$
10y + (10 - 5y) = 15
\Rightarrow 10y + 10 - 5y = 15
\Rightarrow 5y + 10 = 15
\Rightarrow 5y = 5
\Rightarrow y = 1
$$

Then $ x = 10 - 5(1) = 5 $

#### ✔ Check:
- First equation: $ 2(5) + 10(1) = 10 + 10 = 20 ✔️ $
- Second equation: $ 30(1) + 3(5) = 30 + 15 = 45 ✔️ $

Answer: The two numbers are 5 and 1

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✔ Final Answers:

1. 6 dimes and 18 quarters
2. 6 nickels and 8 quarters
3. Loser: 2300 votes, Winner: 2875 votes
4. The two numbers are 5 and 1

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