Question

8 books cost $19.60. which equation would help determine the cost of 10 books? choose 1 answer: a (\frac{x}{10} = \frac{8}{$19.60}) b (\frac{10}{$19.60} = \frac{x}{8}) c (\frac{8}{10} = \frac{x}{$19.60}) d (\frac{10}{8} = \frac{$19.60}{x}) e none of the above

Explanation:

Step1: Understand the proportion

Let \( x \) be the cost of 10 books. The cost per book should be the same. So, the ratio of the number of books to the cost should be constant. For 8 books costing \$19.60, and 10 books costing \( x \), we have the proportion \(\frac{8}{\$19.60}=\frac{10}{x}\) (or cross - multiplied form \(\frac{10}{8}=\frac{\$19.60}{x}\) which is the same as option D). Let's check each option:

• Option A: \(\frac{x}{10}=\frac{8}{\$19.60}\) implies \( x=\frac{8\times10}{\$19.60}\), which is incorrect as it swaps the ratio of books and cost.
• Option B: \(\frac{10}{\$19.60}=\frac{x}{8}\) implies \( x = \frac{10\times8}{\$19.60}\), which is incorrect.
• Option C: \(\frac{8}{10}=\frac{x}{\$19.60}\) implies \( x=\frac{8\times\$19.60}{10}\), which is incorrect.
• Option D: \(\frac{10}{8}=\frac{\$19.60}{x}\) cross - multiplying gives \( 10x=8\times\$19.60\), and \( x=\frac{8\times\$19.60}{10}\) is wrong? Wait, no. Wait, the correct proportion is \(\frac{\text{Number of books}_1}{\text{Cost}_1}=\frac{\text{Number of books}_2}{\text{Cost}_2}\), so \(\frac{8}{\$19.60}=\frac{10}{x}\), which can be rewritten as \(\frac{10}{8}=\frac{x}{\$19.60}\)? No, cross - multiplying \(\frac{8}{\$19.60}=\frac{10}{x}\) gives \( 8x = 10\times\$19.60\), and \(\frac{10}{8}=\frac{x}{\$19.60}\) also gives \( 8x=10\times\$19.60\). Wait, option D is \(\frac{10}{8}=\frac{\$19.60}{x}\), cross - multiplying gives \( 10x = 8\times\$19.60\), which is the same as \( 8x=10\times\$19.60\) (just rearranged). Wait, maybe I made a mistake earlier. Let's re - express:

If 8 books cost \$19.60, the cost per book is \(\frac{\$19.60}{8}\). For 10 books, the cost per book is \(\frac{x}{10}\). Since the cost per book is the same, \(\frac{\$19.60}{8}=\frac{x}{10}\), which can be rewritten as \(\frac{10}{8}=\frac{x}{\$19.60}\)? No, cross - multiplying \(\frac{\$19.60}{8}=\frac{x}{10}\) gives \( 8x = 10\times\$19.60\), and \(\frac{10}{8}=\frac{x}{\$19.60}\) also gives \( 8x = 10\times\$19.60\). Wait, option D is \(\frac{10}{8}=\frac{\$19.60}{x}\), cross - multiplying gives \( 10x=8\times\$19.60\), which is equivalent to \( 8x = 10\times\$19.60\) (by cross - multiplying both sides by \( 8x\) and then dividing). Wait, no, let's do it properly.

The correct proportion is based on the fact that the ratio of the number of books to the cost should be equal. So \(\frac{\text{Number of books}}{\text{Cost}}\) is constant. So \(\frac{8}{\$19.60}=\frac{10}{x}\). Cross - multiplying gives \( 8x=10\times\$19.60\), and \(\frac{10}{8}=\frac{\$19.60}{x}\) cross - multiplying gives \( 10x = 8\times\$19.60\), which is the same equation (just rearranged). So option D is correct.

Answer:

D. \(\frac{10}{8}=\frac{\$19.60}{x}\)