Q6. - Algebra 1

Question

Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.

The girls’ volleyball team is selling T-shirts and pennants to raise money for new uniforms. The team hopes to raise more than \(250.

Item	Price
T-shirt	\)10
Pennant	$4

Which of the following combinations of items sold would meet this goal?

F 16 T-shirts and 20 pennants

G 20 T-shirts and 12 pennants

H 18 T-shirts and 18 pennants

J 15 T-shirts and 25 pennants

Step-by-Step Solution

Answer

The combination of items sold that would meet this goal is 18 T-shirts and 18 pennants.

Therefore, the option H is correct.

Step 1. Write the inequality for the given problem.

It is given that the price of 1 T-shirt is $10 and price of 1 pennant is $4.

Let $x$ and $b$ be the number of T-shirts and pennants respectively.

 

The price of 1 T-shirt is $10. Therefore, the price of $x$ T-shirts is .

 

The price of 1 pennant is $4. Therefore, the price of $y$ pennants is .

 

The total price of T-shirts $x$ and $y$ pennants is $\$\left(10x+4y\right)$.

 

The sum of the money is more than $250.

 

Therefore, the inequality for the given problem is:

$\begin{array}{c}10x+4y>250\\ 2\left(5x+2y\right)>250\\ \frac{2\left(5x+2y\right)}{2}>\frac{250}{2}\\ 5x+2y>125\end{array}$

Step 2. Determine the combination of items sold that would meet this goal.

The inequality for the given problem is:

$5x+2y>125$

If the number of T-shirts is 16 and the number of pennants is 20, that implies if the value of is 16 and the value of  is 20 then it can be observed that:

$\begin{array}{c}5x+2y>125\\ 5\left(16\right)+2\left(20\right)>125\\ 80+40>125\\ 120>125\end{array}$

The condition obtained is $120>125$ which is false, Therefore, the values $x=16$ and $y=20$ does not satisfy the inequality $5x+2y>125$.

 

If the number of T-shirts is 20 and the number of pennants is 12, that implies if the value of $x$ is 20 and the value of $y$ is 12 then it can be observed that:

$\begin{array}{c}5x+2y>125\\ 5\left(20\right)+2\left(12\right)>125\\ 100+24>125\\ 124>125\end{array}$

The condition obtained is $124>125$ which is false, Therefore, the values $x=20$ and $y=12$ does not satisfy the inequality $5x+2y>125$.

 

If the number of T-shirts is 18 and the number of pennants is 18, that implies if the value of $x$ is 18 and the value of $y$ is 18 then it can be observed that:

$\begin{array}{c}5x+2y>125\\ 5\left(18\right)+2\left(18\right)>125\\ 90+36>125\\ 126>125\end{array}$

The condition obtained is $126>125$ which is true, Therefore, the values $x=18$ and $y=18$ satisfies the inequality $5x+2y>125$.

 

If the number of T-shirts is 15 and the number of pennants is 25, that implies if the value of $x$ is 15 and the value of $y$ is 25 then it can be observed that:

$\begin{array}{c}5x+2y>125\\ 5\left(15\right)+2\left(25\right)>125\\ 75+50>125\\ 125>125\end{array}$

The condition obtained is $125>125$ which is false, Therefore, the values $x=15$ and $y=25$ does not satisfy the inequality $5x+2y>125$.

 

As, the values $x=18$ and $y=18$ satisfies the inequality $5x+2y>125$. Therefore, $x=18$ and $y=18$ are the solutions of the inequality $5x+2y>125$.

 

Therefore, the combination of items sold that would meet this goal is 18 T-shirts and 18 pennants.

 

Therefore, the option H is correct.