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Answer :
To solve the problem, we need to determine how many copies of each book there are. Let's tackle this by setting up and solving a system of equations based on the information provided:
1. Define the variables:
- Let [tex]\( x \)[/tex] be the number of copies of the first paperback book.
- Let [tex]\( y \)[/tex] be the number of copies of the second paperback book.
2. Set up the equations based on the problem:
- The total number of books is 179, so our first equation is:
[tex]\[
x + y = 179
\][/tex]
- The total weight of the books is 128 pounds, with each copy of the first book weighing [tex]\(\frac{2}{3}\)[/tex] of a pound and each copy of the second book weighing [tex]\(\frac{3}{4}\)[/tex] of a pound. Therefore, our second equation is:
[tex]\[
\frac{2}{3}x + \frac{3}{4}y = 128
\][/tex]
3. Simplify the second equation:
- To eliminate the fractions, let's multiply the entire equation by 12 (the least common multiple of 3 and 4) to make calculations easier:
[tex]\[
12 \left(\frac{2}{3}x + \frac{3}{4}y\right) = 12 \times 128
\][/tex]
[tex]\[
8x + 9y = 1536
\][/tex]
4. Now we have a system of equations:
[tex]\[
\begin{align*}
x + y &= 179 \\
8x + 9y &= 1536
\end{align*}
\][/tex]
5. Solve the system of equations:
- You can solve this system by either substitution or elimination. For this explanation, I will assume the solution was found using a standard method to find:
[tex]\[
x = 75 \quad \text{and} \quad y = 104
\][/tex]
6. Check the answer:
- The total number of books: [tex]\( 75 + 104 = 179 \)[/tex], which matches the problem's condition.
- The total weight:
[tex]\[
\frac{2}{3} \times 75 + \frac{3}{4} \times 104 = 128
\][/tex]
- Both conditions are satisfied by these values.
Therefore, there are 75 copies of the first paperback book and 104 copies of the second paperback book.
1. Define the variables:
- Let [tex]\( x \)[/tex] be the number of copies of the first paperback book.
- Let [tex]\( y \)[/tex] be the number of copies of the second paperback book.
2. Set up the equations based on the problem:
- The total number of books is 179, so our first equation is:
[tex]\[
x + y = 179
\][/tex]
- The total weight of the books is 128 pounds, with each copy of the first book weighing [tex]\(\frac{2}{3}\)[/tex] of a pound and each copy of the second book weighing [tex]\(\frac{3}{4}\)[/tex] of a pound. Therefore, our second equation is:
[tex]\[
\frac{2}{3}x + \frac{3}{4}y = 128
\][/tex]
3. Simplify the second equation:
- To eliminate the fractions, let's multiply the entire equation by 12 (the least common multiple of 3 and 4) to make calculations easier:
[tex]\[
12 \left(\frac{2}{3}x + \frac{3}{4}y\right) = 12 \times 128
\][/tex]
[tex]\[
8x + 9y = 1536
\][/tex]
4. Now we have a system of equations:
[tex]\[
\begin{align*}
x + y &= 179 \\
8x + 9y &= 1536
\end{align*}
\][/tex]
5. Solve the system of equations:
- You can solve this system by either substitution or elimination. For this explanation, I will assume the solution was found using a standard method to find:
[tex]\[
x = 75 \quad \text{and} \quad y = 104
\][/tex]
6. Check the answer:
- The total number of books: [tex]\( 75 + 104 = 179 \)[/tex], which matches the problem's condition.
- The total weight:
[tex]\[
\frac{2}{3} \times 75 + \frac{3}{4} \times 104 = 128
\][/tex]
- Both conditions are satisfied by these values.
Therefore, there are 75 copies of the first paperback book and 104 copies of the second paperback book.
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