Question

3-60. solve each equation below for x. then check your solutions.
a. \\( \frac{x}{8} = \frac{3}{4} \\)
b. \\( \frac{2}{5} = \frac{x}{40} \\)
c. \\( \frac{1}{8} = \frac{x}{12} \\)
d. \\( \frac{x}{10} = \frac{12}{15} \\)
3-61.

Explanation:

Step1: Solve equation a: $\frac{x}{8}=\frac{3}{4}$

To solve for \( x \), we can cross - multiply. Cross - multiplying gives us \( 4x = 8\times3 \).
Simplify the right - hand side: \( 4x=24 \).
Divide both sides by 4: \( x = \frac{24}{4}=6 \).
Check: Substitute \( x = 6 \) into the original equation. Left - hand side: \( \frac{6}{8}=\frac{3}{4} \), right - hand side: \( \frac{3}{4} \). Since \( \frac{3}{4}=\frac{3}{4} \), the solution is correct.

Step2: Solve equation b: $\frac{2}{5}=\frac{x}{40}$

Cross - multiply: \( 5x=2\times40 \).
Simplify the right - hand side: \( 5x = 80 \).
Divide both sides by 5: \( x=\frac{80}{5}=16 \).
Check: Substitute \( x = 16 \) into the original equation. Left - hand side: \( \frac{2}{5} \), right - hand side: \( \frac{16}{40}=\frac{2}{5} \). Since \( \frac{2}{5}=\frac{2}{5} \), the solution is correct.

Step3: Solve equation c: $\frac{1}{8}=\frac{x}{12}$

Cross - multiply: \( 8x = 1\times12 \).
Simplify the right - hand side: \( 8x=12 \).
Divide both sides by 8: \( x=\frac{12}{8}=\frac{3}{2}=1.5 \).
Check: Substitute \( x = \frac{3}{2} \) into the original equation. Left - hand side: \( \frac{1}{8} \), right - hand side: \( \frac{\frac{3}{2}}{12}=\frac{3}{2}\times\frac{1}{12}=\frac{1}{8} \). Since \( \frac{1}{8}=\frac{1}{8} \), the solution is correct.

Step4: Solve equation d: $\frac{x}{10}=\frac{12}{15}$

First, simplify \( \frac{12}{15}=\frac{4}{5} \). So the equation becomes \( \frac{x}{10}=\frac{4}{5} \).
Cross - multiply: \( 5x = 4\times10 \).
Simplify the right - hand side: \( 5x = 40 \).
Divide both sides by 5: \( x=\frac{40}{5}=8 \).
Check: Substitute \( x = 8 \) into the original equation. Left - hand side: \( \frac{8}{10}=\frac{4}{5} \), right - hand side: \( \frac{12}{15}=\frac{4}{5} \). Since \( \frac{4}{5}=\frac{4}{5} \), the solution is correct.

Answer:

a. \( x = 6 \)

b. \( x = 16 \)

c. \( x=\frac{3}{2}\) (or \( x = 1.5 \))

d. \( x = 8 \)