4.3 Number Analogies: Fractions, Signs, and Solution Audits

Audit the arithmetic and the model separately

A number-analogy solution can fail because the rule is unsupported or because a valid rule was calculated incorrectly. Keep those two checks separate. First ask whether the proposed operation works on both given pairs. Then verify the target calculation, including signs, parentheses, and fraction operations.

These original walkthroughs state the rule where needed so that the calculations have an unambiguous target. They teach techniques transferable to paired-number reasoning without claiming that each technique is an official high-school subtopic or appears at a particular frequency.

Walkthrough: multiply, then subtract

The stated rule is “multiply the input by four, then subtract three.” Known pairs are $(2,5)$ and $(5,17)$. Predict the output for eight.

1. First pair: $4(2)-3=8-3=5$.
2. Second pair: $4(5)-3=20-3=17$.
3. Target: $4(8)-3=32-3=29$.
4. Check order: $(8-3)\times4=20$ would subtract first and therefore use a different rule.

A plausible distractor can be a correctly computed answer to the wrong expression. Explain the expression before judging its arithmetic. Do not describe twenty as a random mistake when it specifically reflects reversed operation order.

Walkthrough: fractions

Use “divide the input by two, then add three.” Inputs four and ten produce five and eight. For input seven, calculate:

$$\frac{7}{2}+3=\frac{7}{2}+\frac{6}{2}=\frac{13}{2}=6.5.$$

The result is fractional, and that is not a reason to change the prompt. A practice rule can legitimately produce fractions. If the options are all integers, inspect your calculation and the item's wording, but do not alter a given number to force an integer answer.

For the distinct rule “add three, then divide by two,” input seven gives $(7+3)/2=5$. The position of the addition matters. Write parentheses when the entire sum is divided.

Walkthrough: reciprocals

A reciprocal of a nonzero number is one divided by that number. If a stated practice rule is $y=1/x+2$, then input two gives $1/2+2=5/2$, and input four gives $1/4+2=9/4$. Input one-half gives $1/(1/2)+2=2+2=4$.

Zero has no reciprocal. If a proposed reciprocal rule must apply to an input of zero, it is undefined there and cannot explain an ordinary finite output for that pair. This is a genuine mathematical restriction, unlike an invented preference for integer answers.

Distinguish the reciprocal of a fraction from division by an integer. The reciprocal of $3/4$ is $4/3$; dividing $3/4$ by two gives $3/8$. State which operation the rule requires.

Walkthrough: signs and squares

Suppose the stated rule is $y=x^2-4$. For input negative three:

$$(-3)^2-4=9-4=5.$$

For the different rule $y=-x^2-4$, the result is $-9-4=-13$. The negative sign outside a square is not part of the squared base. Parentheses communicate that distinction.

A subtraction sign also has direction. “Subtract the input from twenty” means $20-x$, not $x-20$. At input seven the two expressions yield thirteen and negative thirteen. Translate the verbal instruction carefully before calculating.

Test competing candidates

For the pair $(3,14)$, consider these rules: $x^2+5$, $2x+4$, $6x-5$, and $5x-1$. The first and fourth produce fourteen; the second produces ten and the third thirteen. Now add $(6,41)$. The square-plus-five rule also produces forty-one, while $5x-1$ produces twenty-nine. Only the first of these four listed candidates fits both pairs.

This establishes a best candidate among the listed rules. It does not establish that no other mathematical function fits the same two points. If a separate exercise offers both fifty-nine and sixty-nine as target outputs for eight, the linear competitor $9x-13$ must also be considered unless the prompt constrains the model.

Record the exact failure

A useful correction includes the failed calculation and the repaired one. That evidence is more informative than writing “be careful.” Try a new input afterward to verify that the repair transfers.

Build fluency without skipping checks

Practice accurate multiplication, common squares, and fraction equivalences in short exercises. Then mix them with analogy tasks. Faster arithmetic can reduce the work required for a candidate test, but it does not validate an unsupported rule.

The complete standard Number Analogies subtest has 18 items in 10 minutes. Its approximately 33.3-second average is not a required item deadline. Decide how to use your assigned window from actual practice and available navigation, while obeying the proctor's stop instructions. Correct arithmetic, a supported model, and a final direction check remain the core of the solution.