Here are the step-by-step solutions for the four problems on your 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 ($2$ times the number of students, $x$). This equals the prizes left ($3$).
$$31 - 2x = 3$$
Step 2: Solve for x.
First, get the term with $x$ by itself. You can subtract 31 from both sides:
$$-2x = 3 - 31$$
$$-2x = -28$$
Now, divide by -2 to isolate $x$:
$$x = \frac{-28}{-2}$$
$$x = 14$$
Check: If there are 14 students, he gives away $14 \times 2 = 28$ prizes. $31 - 28 = 3$. This matches the problem.
***
Problem 2
The Problem: A family spent $\$52$ total. Tickets cost $\$9.75$ each ($x$ tickets). There was a $\$3.25$ service fee. Find $x$.
Step 1: Write the equation.
The cost of the tickets plus the service fee equals the total spent.
$$9.75x + 3.25 = 52$$
Step 2: Solve for x.
Subtract the service fee ($3.25$) from the total ($52$):
$$9.75x = 52 - 3.25$$
$$9.75x = 48.75$$
Divide by $9.75$ to find the number of tickets:
$$x = \frac{48.75}{9.75}$$
$$x = 5$$
Check: 5 tickets at $\$9.75$ is $\$48.75$. Add the $\$3.25$ fee, and you get $\$52.00$.
***
Problem 3
The Problem: Jarret bought bikes for $\$350$ total. He sold them for $\$50$ each ($x$ bikes). His profit was $\$6200$. Find $x$.
Step 1: Write the equation.
Profit is equal to Money Made (Revenue) minus Money Spent (Cost).
Money Made = $\$50$ per bike $\times$ $x$ bikes ($50x$).
Money Spent = $\$350$.
Profit = $\$6200$.
Equation:
$$50x - 350 = 6200$$
Step 2: Solve for x.
Add $350$ to both sides to isolate the $50x$:
$$50x = 6200 + 350$$
$$50x = 6550$$
Divide by $50$:
$$x = \frac{6550}{50}$$
$$x = 131$$
Check: Selling 131 bikes at $\$50$ makes $\$6550$. Subtract the $\$350$ cost, and the profit is $\$6200$.
***
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.
If he sells $\frac{2}{3}$ of his rugs, he keeps $\frac{1}{3}$ of them.
So, he starts with $\frac{1}{3}x$.
Then he adds 128 rugs.
The total is 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 both sides by 3 to get $x$ by itself:
$$x = 199 \times 3$$
$$x = 597$$
Check: Start with 597 rugs. Sell $\frac{2}{3}$ (which is 398), leaving 199. Buy 128 more ($199 + 128$), which equals 327.
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Final Answer:
1.
14 students
2.
5 tickets
3.
131 bikes
4.
597 rugs
Parent Tip: Review the logic above to help your child master the concept of writing equations from word problems worksheet.
