Q74. - Algebra 2

Question

Allison is saving money to buy a video game system. In the first week, her savings were \(8 less than $\frac{\mathbf{2}}{\mathbf{5}}$ the price of the system. In the second week, she saved 50 cents more than $\frac{\mathbf{1}}{\mathbf{2}}$ the price of the system. She was still \)37 short. Find the price of the system.

Step-by-Step Solution

Answer

The price of the system is $295.

Step 1 ­- Description of step.

Let the price of the system be $$x$.

It is given that the savings in the first week was $8 less than $\frac{2}{5}$ the price of the system.

Therefore, the savings in the first week is given by the expression:

$\frac{2}{5}x-8$

It is also given that the savings in the second week was 50 cents more than $\frac{1}{2}$ the price of the system.

It is known fact that $\text{1 cent = }\frac{\text{1}}{\text{100}}\text{dollars}$.

Therefore, it can be obtained that:

$\begin{array}{c}\text{50 cent = }\frac{\text{50}}{\text{100}}\text{dollars}\\ \text{=}\frac{\text{1}}{\text{2}}\text{dollars}\end{array}$

Now, the savings in the second week is given by the expression:

$\frac{1}{2}x+\frac{1}{2}$

Step 2 ­- Description of step.

It is given that still she was $37 short after saving the money for two weeks.

The savings in the first week is $\frac{2}{5}x-8$ and the savings in the second week is $\frac{1}{2}x+\frac{1}{2}$.

Therefore, the total savings is given by:

$\frac{2}{5}x-8+\frac{1}{2}x+\frac{1}{2}$.

Now, as she was still $37 short , therefore it can be noticed that:

$\frac{2}{5}x-8+\frac{1}{2}x+\frac{1}{2}=x-37$

Step 3 ­- Find the value of x.

Solve the equation $\frac{2}{5}x-8+\frac{1}{2}x+\frac{1}{2}=x-37$ to find the value of $x$.

Therefore, it can be obtained that:

$\begin{array}{c}\frac{2}{5}x-8+\frac{1}{2}x+\frac{1}{2}=x-37\\ 10\left(\frac{2}{5}x-8+\frac{1}{2}x+\frac{1}{2}\right)=10\left(x-37\right)\\ 4x-80+5x+5=10x-370\\ 9x-75=10x-370\\ 9x-75+370=10x-370+370\\ 9x+295=10x\\ 9x-9x+295=10x-9x\\ 295=x\end{array}$

Therefore, the value of $x$ is 295.

Step 4 ­- Write the price of the system.

Therefore, the price of the system is $295.