To find the missing term in a proportion a/b = c/d, cross-multiply to get ad = bc, then divide by the number paired with the unknown. For 12:18 = ?:9, cross-multiplying gives 12 × 9 = 108, and 108 ÷ 18 = 6 — so the missing term is 6.
Proportions run the numbers behind recipes, map scales, medicine doses, and exchange rates: one complete ratio is known, one ratio has a hole in it, and the hole must preserve the same relationship. Two methods solve it — cross-multiplication works in every position, and the unit-rate method is faster when the numbers divide cleanly.
What a Proportion Actually Is
A proportion is a statement that two ratios are equal. 12:18 = 6:9 is a proportion because both ratios reduce to 2:3 — they describe the same relationship. Written as fractions it reads 12/18 = 6/9. The terms even have names: in a:b = c:d, the outer terms a and d are the extremes and the inner terms b and c are the means. That vocabulary matters because the whole method rests on one fact: the product of the extremes equals the product of the means.
Cross-Multiplication: The 4-Step Method
Cross-multiplication is not a trick — it is one algebraic step. Starting from a/b = c/d, multiply both sides by b and then by d: the denominators cancel, leaving a × d = b × c. That single equation solves for any missing term.
Worked example: 12:18 = ?:9.
- Write it with a letter for the unknown: 12/18 = x/9.
- Cross-multiply: 12 × 9 = 18 × x, so 108 = 18x.
- Divide by the number paired with x: x = 108 ÷ 18 = 6.
- Check: 6:9 simplifies to 2:3, exactly like 12:18 — or check the cross products: 12 × 9 = 108 and 18 × 6 = 108 ✓.
Prefer the instant route? Calcrio’s free ratio proportion solver finds any missing term and prints the cross-multiplication steps, so you can compare its working with yours.
When the Missing Term Sits in the Denominator
The method does not care where the hole is. Take 3/4 = 9/x:
- Cross-multiply: 3 × x = 4 × 9, so 3x = 36.
- Divide by 3: x = 12.
- Check: 9/12 simplifies to 3/4 ✓ (9 ÷ 3 = 3, 12 ÷ 3 = 4).
Beginners often assume the unknown must be on top — it doesn’t. The same ad = bc equation isolates whichever term is missing, top or bottom, left or right. This is the exact skill that word problems like “3 out of 4 students passed; how many passed out of 9?” are testing.
The Unit-Rate Method: Faster When Numbers Divide Cleanly
Sometimes you can skip the cross-multiplication. In 12:18 = ?:9, notice that 18 shrinks to 9 — divide by 2. So divide the other side by 2 as well: 12 ÷ 2 = 6. Equivalently, find the unit rate: 12/18 = 2/3 per one unit of the second term, and 2/3 × 9 = 6.
| Method | How it works | Best when |
|---|---|---|
| Cross-multiplication | ad = bc, then divide | The unknown can sit anywhere; always works |
| Unit rate / factor of change | Spot the multiply-or-divide link between the known pair | Numbers divide cleanly (like 18 → 9) |
Cross-multiplication is the safe default — it never depends on spotting a pattern. The unit rate is the shortcut for friendly numbers. When the link isn’t obvious (try 7/13 = x/29 by inspection), cross-multiply.
The Order Trap: Keep Comparisons Aligned
The single biggest source of wrong answers is mismatched order. Take a recipe: 2 cups of flour to 3 cups of sugar. How much flour for 15 cups of sugar? Both sides must read flour-to-sugar: 2/3 = x/15. Cross-multiply: 3x = 30, so x = 10 cups of flour ✓ (2:3 = 10:15, both reduce to 2:3).
Flip one side and the answer breaks: 3/2 = x/15 gives x = 22.5 — a ruined cake. The rule is simple: the units must occupy the same positions on both sides. If the left side reads flour over sugar, the right side must read flour over sugar. Write the words above each number before you write the numbers themselves, and the trap disappears.
3 Mistakes That Cost Marks
1. Cross-adding instead of cross-multiplying. For 12/18 = x/9, adding diagonally gives 21 = 18 + x, which is meaningless. Only the products matter: 12 × 9 and 18 × x.
2. Mismatched comparison order. As the recipe example shows, flipping one ratio silently changes the question. Align the units first, calculate second.
3. Rounding the unit rate early. If the unit rate doesn’t divide evenly (say 7/13 ≈ 0.538), rounding it and multiplying back introduces error. Keep the fraction exact — or just cross-multiply, which stays exact to the end.
Frequently Asked Questions
What is the missing term in 12:18 = ?:9?
6. Cross-multiply: 12 × 9 = 108, and 108 ÷ 18 = 6. Check: 6:9 simplifies to 2:3, the same as 12:18.
What is cross multiplication?
The rule that if a/b = c/d, then a × d = b × c. It comes from multiplying both sides of the proportion by both denominators so they cancel. From that equation you can solve for any unknown term.
Can the missing term be in the denominator?
Yes. In 3/4 = 9/x, cross-multiplying gives 3x = 36, so x = 12. The method works wherever the unknown sits — top or bottom, left or right.
How do you check a proportion?
Compare the cross products: if a/b = c/d is true, then a × d equals b × c. For 12:18 = 6:9, both products are 108, so the proportion holds.
What’s the difference between a ratio and a proportion?
A ratio compares two quantities (12:18). A proportion states that two ratios are equal (12:18 = 6:9). You solve a proportion whenever one of its four terms is unknown.
Keep Practicing
Work these by hand and verify with the free ratio calculator: 4:7 = x:21 gives x = 12 (4 × 21 = 84, 84 ÷ 7 = 12); 5:x = 15:36 gives x = 12 (5 × 36 = 180, 180 ÷ 15 = 12); and 2:5 = 8:x gives x = 20. Fractions obey the same rules — the fraction calculator shows the identical cross-multiplication step when you divide fractions.