xy1431252040kIn the table above, the ratio of y to x for each ordered pair is constant. What is the value...

1. TRANSLATE the problem information

• Given information:
  • Table with ordered pairs (x,y)
  • The ratio of y to x is constant for each pair
  • Need to find k when x = 40

• What this tells us: \(\mathrm{y/x}\) should equal the same number for every row

2. INFER the approach

• Since the ratio is constant, I can use any complete pair to find what that constant is
• Once I know the constant ratio, I can use it to find k

3. SIMPLIFY to find the constant ratio

• Using the first pair (1,4): \(\mathrm{y/x = 4/1 = 4}\)
• Let me verify with another pair (3,12): \(\mathrm{y/x = 12/3 = 4}\) ✓
• The constant ratio is 4

4. TRANSLATE this to find k

• For the pair (40,k), the ratio must also be 4
• So \(\mathrm{k/40 = 4}\)

5. SIMPLIFY to solve for k

• \(\mathrm{k/40 = 4}\)
• \(\mathrm{k = 4 \times 40 = 160}\)

Answer: D. 160

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misunderstand "ratio of y to x" and instead calculate \(\mathrm{x/y}\) ratios, getting \(\mathrm{1/4}\) instead of 4.

Using \(\mathrm{x/y = 1/4}\), they would set up \(\mathrm{40/k = 1/4}\), leading to \(\mathrm{k = 40 \times 4 = 160}\). Interestingly, this still gives the right answer due to the mathematical relationship, but it represents conceptual confusion about ratio direction.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the ratio as 4 but make arithmetic errors in the final calculation.

For example, they might confuse multiplication and division: instead of \(\mathrm{k = 4 \times 40}\), they calculate \(\mathrm{k = 40 \div 4 = 10}\), or make other calculation mistakes. Since 10 isn't an answer choice, this leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can correctly interpret ratio language and consistently apply proportional relationships. The key insight is recognizing that "constant ratio" creates a direct relationship that can be used predictively.