To solve this problem, let's analyze the effect of changing the length and width of a rectangle on its area. Here's a step-by-step solution:
1. Identify the Ratios:
- The length of the rectangle is increased in the ratio \( \frac{7}{3} \).
- The width of the rectangle is increased in the ratio \( \frac{2}{5} \).
2. Original and New Dimensions:
- Let the original length of the rectangle be \( L \).
- Let the original width of the rectangle be \( W \).
- The new length of the rectangle becomes \( L \times \frac{7}{3} \).
- The new width of the rectangle becomes \( W \times \frac{2}{5} \).
3. Calculate the New Area:
- The area of the original rectangle is \( A_{original} = L \times W \).
- The area of the new rectangle is:
[tex]\[
A_{new} = \left( L \times \frac{7}{3} \right) \times \left( W \times \frac{2}{5} \right)
\][/tex]
4. Simplify the Area Calculation:
- Simplify the new area expression:
[tex]\[
A_{new} = L \times W \times \frac{7}{3} \times \frac{2}{5}
\][/tex]
- This simplifies to:
[tex]\[
A_{new} = L \times W \times \frac{14}{15}
\][/tex]
5. Determine the Area Ratio:
- The ratio of the new area to the original area is given by:
[tex]\[
\text{Area Ratio} = \frac{A_{new}}{A_{original}} = \frac{L \times W \times \frac{14}{15}}{L \times W}
\][/tex]
- Simplifying the fraction, we get:
[tex]\[
\text{Area Ratio} = \frac{14}{15}
\][/tex]
6. Interpret the Area Ratio:
- The ratio \( \frac{14}{15} \) is approximately \( 0.9333 \).
Thus, when the length of the rectangle is increased in the ratio 7:3 and the width increased in the ratio 2:5, the area of the rectangle actually decreases to approximately 93.33% of its original area.
So, the area of the rectangle is not increased; it is decreased in the ratio [tex]\( 14:15 \)[/tex].
