A car travels 240 miles on 8 gallons of gas. A student sets up the proportion 240/8 = x/360 to find how many gallons are needed for 360 miles. What is wrong with this setup?
ANothing — this proportion is correct and will give the right answer
BThe types are inconsistent: the left side is miles/gallons, but the right side places the unknown over miles — the same type of quantity must appear in the same position on both sides
CCross-multiplication is the wrong method for distance-fuel problems
DThe unknown x should always appear in the numerator on the right side of a proportion
A valid proportion requires the same type of ratio on both sides. 240/8 is miles per gallon; x/360 places x (gallons) over miles — an inverted and inconsistent relationship. Cross-multiplying gives 240 × 360 = 8x → x = 10,800, which is wildly wrong. The correct setup is 240/8 = 360/x (miles/gallons on both sides), giving 240x = 2,880 → x = 12 gallons.
Question 2 Multiple Choice
After correctly cross-multiplying the proportion 3/4 = 9/x, a student writes '3x = 36' but then gives the final answer as x = 33. Which error did they make?
AThey used the wrong cross-multiplication pairs
BThey subtracted 3 from both sides instead of dividing both sides by 3
CCross-multiplying a proportion with a variable in the denominator requires a different technique
DThey should have used the scale-factor method instead of cross-multiplication
After cross-multiplying, 3x = 36 is a one-step equation: divide both sides by 3 to get x = 12. Subtracting 3 gives 33 — treating it as 3 + x = 36 instead of 3 × x = 36. This is a common error among students who haven't internalized that cross-multiplication produces a multiplication equation, not an addition one.
Question 3 True / False
Cross-multiplication works because it transforms a proportion (two equal fractions) into a simple linear equation that can be solved with one-step equation techniques.
TTrue
FFalse
Answer: True
Exactly right. Cross-multiplying a/b = c/d by both denominators (b and d) gives a × d = b × c — an ordinary multiplication equation. This connects proportion-solving to the equation-solving skills you already have, rather than being a separate procedure to memorize.
Question 4 True / False
Setting up the proportion correctly is usually the easy part of solving proportion problems; most errors occur during the cross-multiplication step.
TTrue
FFalse
Answer: False
This common assumption is backwards. Cross-multiplication is mechanical and easy to check. Setup — correctly identifying which quantities correspond and placing them consistently — is where most errors originate. A correct setup with a small arithmetic error is easily caught; an incorrect setup produces a wrong answer that looks perfectly reasonable and goes undetected.
Question 5 Short Answer
Why is it important to keep the same type of quantity in the same position (numerator or denominator) on both sides of a proportion?
Think about your answer, then reveal below.
Model answer: A proportion asserts that two ratios are equal. If the types are inconsistent — miles on top on one side, hours on top on the other — the fractions represent different relationships, and setting them equal is meaningless. Flipping one fraction inverts the ratio, causing x to come out as its reciprocal. For example, 1/50 = x/3 (inches over miles vs. x over inches) gives x = 3/50 miles instead of the correct 150 miles.
The rule 'same type in same position' is the setup equivalent of the calculation rule 'cross-multiply correctly.' Both are necessary. A proportion is only valid when both fractions represent the same relationship — e.g., both are inches per mile, or both are miles per gallon. Violating this produces an equation that's algebraically solvable but meaninglessly wrong.