For the values j and k, the ratio of j to k is 11:12. If j is multiplied by 17,...

1. TRANSLATE the problem information

• Given information:
  • Original ratio: j to k is 11 to 12, which means \(\mathrm{j/k = 11/12}\)
  • j gets multiplied by 17, so new j-value = \(\mathrm{17j}\)
  • Need to find: what k gets multiplied by to keep the same ratio

2. INFER the key relationship

• The crucial insight: When you want to maintain a ratio, if you scale one part by a certain factor, you must scale the other part by the exact same factor
• Think of it like a recipe - if you double one ingredient, you must double all ingredients to maintain the same taste

3. Set up the equation for maintaining the ratio

• Let x = the factor that k gets multiplied by
• New ratio must equal original ratio: \(\mathrm{(17j)/(xk) = j/k}\)

4. SIMPLIFY to find the scaling factor

• From \(\mathrm{(17j)/(xk) = j/k}\)
• Divide both sides by \(\mathrm{j/k}\): \(\mathrm{(17j)/(xk) \div (j/k) = 1}\)
• This gives us: \(\mathrm{17/x = 1}\)
• Therefore: \(\mathrm{x = 17}\)

Answer: 17

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students often think the scaling factor for k should somehow involve the original ratio numbers (11 and 12), not recognizing that ratio maintenance requires identical scaling of both terms.

This misconception might lead them to calculate something like \(\mathrm{17 \times 12/11}\) or try to use the ratio values in some other way, potentially selecting a wrong numerical answer or getting confused and guessing.

The Bottom Line:

The key insight is that ratios are about the relationship between quantities, not their individual values. When that relationship needs to stay constant, both quantities must change by the same multiplicative factor - it's that simple, but students often overthink it by trying to incorporate the original ratio numbers into their scaling calculation.