Proportions & Cross-Multiplication - Sixth Grade Mathematics
A proportion is a mathematical statement asserting that two ratios are completely equal, unlocking algebraic methods for solving missing value problems.
The Formal Definition of a Proportion
An equation stating that two ratios are equal is called a proportion.
Proportion Format:
a / b = c / d or a : b = c : d
(Read: "a is to b as c is to d", where b, d are not 0)
The Extremes and the Means
In the proportion a/b = c/d (written in colon notation as a : b = c : d), the outer terms a and d are called the extremes, while the middle terms b and c are called the means.
Extremes: a and d
Means: b and c
The Cross-Products Property
The Cross-Products Property is one of the most powerful algebraic theorems in mathematics. It states that in any true proportion, the product of the extremes is equal to the product of the means.
The Cross-Products Property:
If a / b = c / d, then a * d = b * c.
Testing Proportionality:
Is 4/6 equal to 10/15?
Cross-Product 1: 4 * 15 = 60
Cross-Product 2: 6 * 10 = 60
Since 60 = 60, the proportion 4/6 = 10/15 is TRUE!
Solving Proportions for an Unknown Variable
When one term in a proportion is unknown (represented by a variable x), you can solve for x using a three-step protocol.
+-------------------------------------------------------------------------+
| SOLVING PROPORTIONS USING CROSS-MULTIPLICATION |
| |
| Step 1: Set up the proportion with aligned units on both sides: |
| Unit A / Unit B = Unit A / Unit B. |
| Step 2: Cross-multiply: Multiply the numerator of each ratio by the |
| denominator of the opposite ratio: a * x = b * c. |
| Step 3: Solve the one-step multiplication equation by dividing both |
| sides by the coefficient of x: x = (b * c) / a. |
+-------------------------------------------------------------------------+
Worked Example: Solve for x: 5 / 8 = x / 40
Step 1: Cross-multiply:
5 * 40 = 8 * x
200 = 8x
Step 2: Divide both sides by 8:
x = 200 / 8
Step 3: Compute:
x = 25.
Verification: 5/8 = 25/40 (5 * 5 = 25, 8 * 5 = 40; Confirmed!).
Setting Up Proportions from Word Problems
The most critical step when solving real-world word problems is maintaining unit consistency across both ratios.
CORRECT UNIT ALIGNMENT: INCORRECT UNIT ALIGNMENT:
Miles Miles Miles Hours
------- = ------- ------- = -------
Hours Hours Hours Miles
(Units match across top and bottom!) (Units are inverted! WRONG!)
Exercises
Exercise 1. Determine whether each pair of ratios forms a true proportion: (a) 6/9 and 10/15; (b) 8/14 and 12/20; (c) 15/25 and 9/15.
Exercise 2. Solve for the unknown variable in each proportion: (a) 3/7 = x/35; (b) n/12 = 15/20; (c) 8/y = 6/15.
Exercise 3. A photograph measuring 4 inches wide and 6 inches tall is enlarged so that its new width is 10 inches. What is the new height of the enlarged photograph?
Solutions
Solution to Exercise 1. For (a) 6/9 and 10/15: cross-multiply: 6 * 15 = 90; 9 * 10 = 90. Since 90 = 90, they form a true proportion. For (b) 8/14 and 12/20: cross-multiply: 8 * 20 = 160; 14 * 12 = 168. Since 160 does not equal 168, they do not form a true proportion. For (c) 15/25 and 9/15: cross-multiply: 15 * 15 = 225; 25 * 9 = 225. Since 225 = 225, they form a true proportion.
Solution to Exercise 2. For (a) 3/7 = x/35: cross-multiply: 7x = 3 * 35 = 105; divide by 7: x = 105 / 7 = 15. For (b) n/12 = 15/20: cross-multiply: 20n = 12 * 15 = 180; divide by 20: n = 180 / 20 = 9. For (c) 8/y = 6/15: cross-multiply: 6y = 8 * 15 = 120; divide by 6: y = 120 / 6 = 20.
Solution to Exercise 3. Set up the proportion keeping Width in the numerator and Height in the denominator: Original Width / Original Height = Enlarged Width / Enlarged Height. Substitute given values: 4 / 6 = 10 / h. Cross-multiply: 4 * h = 6 * 10, so 4h = 60. Divide both sides by 4: h = 60 / 4 = 15. The new height of the enlarged photograph is 15 inches.