To explain why Sonny got [tex]\(240 = 240\)[/tex] after substituting [tex]\( x = 5 \)[/tex] into the proportion [tex]\(\frac{16}{x} = \frac{48}{15}\)[/tex], we need to understand the concept of cross-multiplication and the properties of proportions.
Here is a step-by-step explanation:
1. Substitution:
Sonny substitutes [tex]\( x = 5 \)[/tex] into the proportion [tex]\(\frac{16}{x} = \frac{48}{15}\)[/tex]. This gives us:
[tex]\[
\frac{16}{5} = \frac{48}{15}
\][/tex]
2. Cross Multiplication:
To verify the equality, Sonny cross multiplied the terms of the proportion. Cross multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. Specifically, we do the following multiplications:
- Multiply 16 (the numerator of the first fraction) by 15 (the denominator of the second fraction),
- Multiply 48 (the numerator of the second fraction) by 5 (the substituted value for [tex]\( x \)[/tex]) (the denominator of the first fraction).
Thus, we calculate:
[tex]\[
16 \times 15 = 240
\][/tex]
and
[tex]\[
48 \times 5 = 240
\][/tex]
3. Equality of Cross Products:
After performing the cross multiplications, Sonny found:
[tex]\[
240 = 240
\][/tex]
This shows that both cross products are equal.
4. Conclusion:
The reason Sonny concluded that [tex]\(240 = 240\)[/tex] is because the cross products of the proportion are equal. When the cross products are equal, the original proportion is valid, and the substitution satisfies the condition.
Hence, the correct reason is:
[tex]\[
\text{because the cross products of the proportion are equal}
\][/tex]
