Enlarge a photo and its width-to-height ratio does not change. The statement that the small ratio equals the big one — 2:3 = 4:6 — is a proportion.
In Lesson 4.1 you learned to compare two quantities with a ratio — three cups of flour to two cups of sugar, 3:2. You also saw that a ratio can be scaled up or down and still describe the same mix, so 3:2 and 6:4 are equivalent. A proportion is what you get when you take two ratios that are equal and write that equality down. That one little equals sign turns out to be a powerful machine.
By the end of this lesson you will be able to do five things: say exactly what a proportion means; use the cross-product property (a·d = b·c) to test whether two ratios are really equal; solve a proportion for a missing fourth term; and read a map scale as a proportion to convert between map distance and real distance. We keep one steady habit of color throughout: the first quantity in each ratio is amber, the second quantity is blue, and when a third term appears it is purple. "Increase" reads green; "decrease" reads orange-red.
4.2.1 What a proportion means
Start with the photo above. The small one is 2 wide and 3 tall; enlarging it ×2 makes it 4 wide and 6 tall. The width-to-height ratio of the small photo is 2:3, and of the big one 4:6. Those two ratios are equal — that is exactly why the enlarged photo looks like the original and not stretched or squished. When two ratios are equal we write
2:3=4:6read: "2 is to 3 as 4 is to 6"
and we call that statement a proportion. A ratio by itself is just a comparison; a proportion is a sentence claiming that two comparisons match. Because a ratio can always be written as a fraction, the very same proportion can be written with a fraction bar:
2:3 = 4:6
means the same as
23 = 46
Both forms say 2 compared to 3 is the same comparison as 4 compared to 6. The general pattern, using letters for "any numbers," is
a:b=c:d or ab = cd
An endless chain of equivalent ratios. Any two links you pick form a proportion, because they are all equal: 2:3 = 4:6 = 6:9 = 8:12.
What a proportion is
A proportion is a statement that two ratios are equal: a:b = c:d, also written ab = cd. You can build a proportion out of any two equivalent ratios, and you can scale a ratio up or down by multiplying (or dividing) both terms by the same number.
A proportion needs the equals sign
Be careful with words. A single comparison like 2:3 is a ratio, not a proportion. It only becomes a proportion when you set it equal to another ratio: 2:3 = 4:6. Ratio = one comparison; proportion = two ratios joined by "=".
🎮 Try itIs it a proportion? Test two ratios
Set two ratios a:b and c:d. The widget reduces each ratio to its simplest value and tells you whether the two are equal — that is, whether they form a proportion.
Ratio 1 a2:b3
Ratio 2 c4:d6
4.2.2 The basic property: cross products are equal
Reducing each ratio to compare them works, but it can be slow. There is a faster, exact test that works every time. Take a proportion written as fractions:
ab = cd
Multiply both sides by b and by d — that is, multiply both sides by b·d. On the left the b's cancel; on the right the d's cancel:
ab · b·d
=
cd · b·d a·d = b·c
The two products a·d and b·c are called the cross products, because they multiply across the equals sign on a diagonal — top-left with bottom-right, and top-right with bottom-left. The result is the single most useful fact in this whole chapter:
The cross multiply on the diagonals: top-left times bottom-right (a · d) equals top-right times bottom-left (b · c). If the proportion is true, these two products are equal.
The cross-product property
The proportion ab = cd is true exactly when its cross products are equal:
a·d = b·c. To test any two ratios, multiply across both diagonals and check whether you get the same number.
Think of the fraction bar as a balance scale. The proportion is "balanced" only when the two cross products weigh the same. Multiply the diagonals; if the two totals match, the scale is level and you truly have a proportion. If they differ, the scale tips and the ratios are not equal.
Worked example — testing two ratios
Is 3:4 = 9:12? Cross-multiply: 3·12 = 36 and 4·9 = 36. The cross products match, so yes — it is a true proportion. ✓
Is 2:5 = 3:7? Cross-multiply: 2·7 = 14 and 5·3 = 15. Since 14 ≠ 15, the cross products differ, so no — these ratios are not equal and do not form a proportion. ✗
🎮 Try itSee the cross products as areas
For ab = cd, the two cross products are drawn as rectangles whose areas are a·d and b·c. When the rectangles match in area, you have a true proportion.
a2b3
c4d6
4.2.3 Solving a proportion for an unknown
Here is where the cross-product property earns its keep. Suppose a proportion has three terms you know and one you do not — call the unknown x. Because the cross products must be equal, you get a simple equation with only one unknown, and you can solve it.
Take 34 = x20. Cross-multiply: the diagonal 3·20 must equal the diagonal 4·x. So
3 · 20 = 4 · x
→
60 = 4x
→
x = 604 = 15
A short way to remember it: x equals the product of the two terms on the unknown's diagonal opposite, divided by the lone known term beside x. In ab = xd the cross products give a·d = b·x, so x = a·d ÷ b.
Worked example — x in a different position
Solve 5x = 824. Here x is a denominator. Cross-multiply on the diagonals: 5·24 = x·8. So 120 = 8x, which gives x = 120 ÷ 8 = 15.
Check by cross-multiplying the finished proportion 515 = 824: 5·24 = 120 and 15·8 = 120. They match. ✓
The answer need not be a whole number
Cross-multiplying always works, but sometimes the division does not come out even. 34 = x10 gives 3·10 = 4x, so x = 30 ÷ 4 = 7.5 (that is 152). A non-whole answer is perfectly fine — just leave it as a fraction or a decimal.
🎮 Try itSolve a proportion for x
Choose which slot holds the unknown x, set the three known terms, and watch the cross-multiplication solve it step by step. Non-whole answers appear as a fraction and a decimal.
x is the
4.2.4 Scale on a map (and scale drawings)
A map is a scale drawing of the real world: every length is shrunk by the same factor, so the map and the land form a giant proportion. A map's scale is a ratio, map distance : real distance. You might see it written 1 cm:100 m — "one centimeter on the map stands for one hundred meters on the ground."
Convert to the same unit first
A scale is only a clean ratio of pure numbers when both sides use the same unit. To turn 1 cm:100 m into a number ratio, change 100 m into centimeters: 100 m = 100 × 100 cm = 10000 cm. So the scale is 1:10000. Always match units before you cross-multiply.
At scale 1:10000, a road measuring 3.5 cm on the map is 3.5 × 10000 = 35000 cm = 350 m in real life. Map and ground form a proportion.
To go from map to ground, set up a proportion with the scale on one side and your measurement on the other, then cross-multiply. With scale 1:10000 and a map distance of 3.5 cm:
110000 = 3.5x
→
1·x = 10000·3.5
→
x = 35000 cm = 350 m
Going the other way — from a real distance back to a map distance — uses the same proportion; you just solve for the other letter. The scale is the bridge in both directions.
Worked example — real distance back to map
On the same 1:10000 map, how long is a 2 km road? First match units: 2 km = 200000 cm. Then 110000 = x200000, so 1·200000 = 10000·x, giving x = 200000 ÷ 10000 = 20 cm on the map.
🎮 Try itMap distance to real distance
Pick a map scale and a map distance in centimeters. The widget sets up the proportion, cross-multiplies, and reports the real distance in sensible units (meters, or kilometers when it gets large).
Scale
Map distance (cm)3.5
★ The big ideas, in one breath
A proportion is the statement that two ratios are equal, a:b = c:d, which you can also write as ab = cd. Its key property is that the cross products are equal, a·d = b·c — use it to test two ratios, and to solve for a missing fourth term by dividing. And a map scale is just a ratio of map distance to real distance (matched to the same unit first), so any map question is a proportion you can cross-multiply.
Coming up next — 4.3
You now know what it means for two ratios to be equal. Next you will meet two ways a pair of quantities can travel together: direct proportion, where doubling one doubles the other, and inverse proportion, where doubling one halves the other. The cross-product tool you just built is exactly what powers both.
✎ Exercises 4.2
Work each one out first, then open the answer to check your thinking.
Which of these ratios is equivalent to 2:3? 4:6, 3:4, 6:9, 5:6.
Show answer
4:6 and 6:9. Multiply 2:3 by 2 to get 4:6 and by 3 to get 6:9. (3:4 and 5:6 are not equal to 2:3.)
Use cross products to test: is 6:8 = 9:12 a true proportion?
Show answer
Yes. Cross-multiply: 6·12 = 72 and 8·9 = 72. The cross products match, so it is a proportion.
Test: is 4:7 = 9:16 a proportion?
Show answer
No.4·16 = 64 but 7·9 = 63. Since 64 ≠ 63, the ratios are not equal — close, but not a proportion.
Solve for x: 45 = x35.
Show answer
Cross-multiply: 4·35 = 5·x, so 140 = 5x and x = 28. Check: 4·35 = 140 and 28·5 = 140. ✓
Solve for x (x is a denominator): 9x = 38.
Show answer
Cross-multiply: 9·8 = x·3, so 72 = 3x and x = 24. Check: 9/24 = 3/8 because 9·8 = 72 and 24·3 = 72. ✓
Solve for x and leave a non-whole answer as a fraction and a decimal: 25 = x9.
Show answer
Cross-multiply: 2·9 = 5·x, so 18 = 5x and x = 18/5 = 3.6. A non-whole answer is fine.
Map → ground. A trail is 6 cm long on a 1:25000 map. How long is it in real life, in kilometers?
Show answer
Real length = 6 × 25000 = 150000 cm. Convert: 150000 cm = 1500 m = 1.5 km.
Ground → map. Two towns are 8 km apart. On a 1:200000 map, how many centimeters apart are they?
Show answer
Match units: 8 km = 800000 cm. Then 1200000 = x800000, so x = 800000 ÷ 200000 = 4 cm.
Recipe. A pancake recipe uses flour to milk in the ratio 3:2 (cups). You pour in 12 cups of flour. How much milk keeps the same proportion?
Show answer
Set up 32 = 12x. Cross-multiply: 3·x = 2·12 = 24, so x = 8 cups of milk. (The recipe scaled up ×4.)
Similar rectangles. A small rectangle is 5 cm by 8 cm. A larger, similar rectangle has the same width-to-height ratio and is 15 cm wide. How tall is it? Then explain why "just add 10 to each side" would be wrong.
Show answer
Similar means equal ratios: 58 = 15x. Cross-multiply: 5·x = 8·15 = 120, so x = 24 cm. Adding 10 to each side would give 15 by 18, but 15:18 reduces to 5:6, not 5:8 — so the shape would be distorted. Ratios scale by multiplying, not by adding.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson develops proportional reasoning and is aligned to the U.S. Common Core standards 6.RP.A.3 (use ratio and rate reasoning, including tables of equivalent ratios), 7.RP.A.2 (recognize and represent proportional relationships between quantities), and 7.G.A.1 (solve problems involving scale drawings, computing actual lengths from a scale). The deliberate emphasis is on the meaning of equality between ratios first, then the cross-product property as both a test and a solving tool, and finally map scale as a real-world proportion. The single most common error — addressed directly in Section 4.2.1, the similar-rectangles exercise, and the quiz — is treating a ratio additively (adding the same amount to both terms) instead of multiplicatively (scaling both terms by the same factor); the antidote students should rehearse is "to keep a ratio, multiply both terms by the same number, then check with cross products."