Question

Use cross multiplication to decide whether the proportion is true or false.        
StartFraction 5 and one fifth Over 1 and one half EndFraction equals StartFraction 31 and one fifth Over 7 and one half EndFraction
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Part 1
Is the proportion true or​ false?

Answer 1

To determine whether the proportion

\[
\frac{5 \frac{1}{5}}{1 \frac{1}{2}} = \frac{31 \frac{1}{5}}{7 \frac{1}{2}}
\]

is true or false, we will first convert all mixed numbers to improper fractions.

1. Convert \(5 \frac{1}{5}\) to an improper fraction:
\[
5 \frac{1}{5} = 5 \times 5 + 1 = 25 + 1 = 26 \quad \Rightarrow \quad \frac{26}{5}
\]

2. Convert \(1 \frac{1}{2}\) to an improper fraction:
\[
1 \frac{1}{2} = 1 \times 2 + 1 = 2 + 1 = 3 \quad \Rightarrow \quad \frac{3}{2}
\]

3. Convert \(31 \frac{1}{5}\) to an improper fraction:
\[
31 \frac{1}{5} = 31 \times 5 + 1 = 155 + 1 = 156 \quad \Rightarrow \quad \frac{156}{5}
\]

4. Convert \(7 \frac{1}{2}\) to an improper fraction:
\[
7 \frac{1}{2} = 7 \times 2 + 1 = 14 + 1 = 15 \quad \Rightarrow \quad \frac{15}{2}
\]

Now we can rewrite the proportion:

\[
\frac{\frac{26}{5}}{\frac{3}{2}} = \frac{\frac{156}{5}}{\frac{15}{2}}
\]

Next, we can cross-multiply to determine if the two sides are equal:

Left side:
\[
26 \cdot 2 = 52
\]
Right side:
\[
156 \cdot 5 = 780
\]

Now we compare both results:
- From the left side: \(52\)
- From the right side: \(780\)

Since \(52 \neq 780\), the proportion is **false**.