Here are the step-by-step solutions for each problem on the worksheet.
Problem 1
The Problem: Eduardo won 31 prizes. He gave 2 prizes to every student in his class ($x$). He has 3 prizes left. Find $x$.
Step 1: Write the equation.
* Start with the total prizes:
31
* Subtract the prizes given away: He gave
2 to
$x$ students, so that is
$2x$.
* The result is what is left:
3
* Equation: $31 - 2x = 3$
Step 2: Solve for $x$.
* Subtract 31 from both sides:
$-2x = 3 - 31$
$-2x = -28$
* Divide by -2:
$x = \frac{-28}{-2}$
$x = 14$
Check: If there are 14 students, he gives away $14 \times 2 = 28$ prizes. $31 - 28 = 3$ left. This is correct.
***
Problem 2
The Problem: A family spent $\$52$ total. Tickets were $\$9.75$ each ($x$ tickets). There was a $\$3.25$ service fee. Find $x$.
Step 1: Write the equation.
* Cost of tickets:
$9.75x$
* Plus the fee:
$+ 3.25$
* Equals total cost:
$= 52$
* Equation: $9.75x + 3.25 = 52$
Step 2: Solve for $x$.
* Subtract 3.25 from both sides:
$9.75x = 52 - 3.25$
$9.75x = 48.75$
* Divide by 9.75:
$x = \frac{48.75}{9.75}$
$x = 5$
Check: 5 tickets at $\$9.75$ is $\$48.75$. Add the $\$3.25$ fee, and you get $\$52.00$. This is correct.
***
Problem 3
The Problem: Jarret bought bikes for $\$350$ total. He sold them for $\$50$ each ($x$ bikes). He made a $\$6200$ profit. Find $x$.
Step 1: Write the equation.
* Profit is calculated as: (Money earned from sales) minus (Cost to buy).
* Money earned:
$50x$
* Cost to buy:
$350$
* Profit:
$6200$
* Equation: $50x - 350 = 6200$
Step 2: Solve for $x$.
* Add 350 to both sides:
$50x = 6200 + 350$
$50x = 6550$
* Divide by 50:
$x = \frac{6550}{50}$
$x = 131$
Check: Selling 131 bikes at $\$50$ earns $\$6550$. Subtract the $\$350$ cost, and the profit is $\$6200$. This is correct.
***
Problem 4
The Problem: Stanley sells $\frac{2}{3}$ of his rugs ($x$). Then he buys 128 more. He now has 327 rugs. Find $x$ (the starting number).
Step 1: Write the equation.
* Start with
$x$ rugs.
* He sells $\frac{2}{3}$ of them. This means he keeps $\frac{1}{3}$ of them. So, he has
$\frac{1}{3}x$ left.
*(Note: You can also write this as $x - \frac{2}{3}x$, which simplifies to $\frac{1}{3}x$)*
* He buys 128 more:
$+ 128$
* Total now:
$= 327$
* Equation: $\frac{1}{3}x + 128 = 327$
Step 2: Solve for $x$.
* Subtract 128 from both sides:
$\frac{1}{3}x = 327 - 128$
$\frac{1}{3}x = 199$
* Multiply by 3 to isolate $x$:
$x = 199 \times 3$
$x = 597$
Check: Start with 597. Sell $\frac{2}{3}$ (which is 398). You have 199 left. Buy 128 more ($199 + 128$). You have 327. This is correct.
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Final Answer:
1. x = 14
2. x = 5
3. x = 131
4. x = 597
Parent Tip: Review the logic above to help your child master the concept of word problems worksheet 7th grade.
