What is Comparing Fractions Using Cross-Multiplication?

S1-SA1-1228

Grade Level:

Class 5

Maths, Computing, AI, Algorithms

Definition

What is it?

Comparing fractions using cross-multiplication is a simple method to find out which of two fractions is bigger or smaller without converting them to decimals or finding a common denominator. You multiply the numerator of one fraction by the denominator of the other, and then compare these two products.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, eating a pizza. Rohan ate 3/5 of his pizza, and Priya ate 2/3 of her pizza. To see who ate more, we can use cross-multiplication. We multiply 3 (Rohan's numerator) by 3 (Priya's denominator) to get 9. Then we multiply 2 (Priya's numerator) by 5 (Rohan's denominator) to get 10. Since 10 is greater than 9, Priya ate more pizza.

Worked Example

Step-by-Step

Let's compare the fractions 2/7 and 3/8.

Step 1: Write down the two fractions you want to compare: 2/7 and 3/8.

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Step 2: Multiply the numerator of the first fraction (2) by the denominator of the second fraction (8). This gives 2 * 8 = 16.

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Step 3: Write this product (16) above the first fraction (2/7).

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Step 4: Multiply the numerator of the second fraction (3) by the denominator of the first fraction (7). This gives 3 * 7 = 21.

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Step 5: Write this product (21) above the second fraction (3/8).

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Step 6: Now compare the two products: 16 and 21. Since 16 is less than 21 (16 < 21).

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Step 7: This means the first fraction is smaller than the second fraction.

Answer: So, 2/7 < 3/8.

Why It Matters

Understanding how to compare fractions efficiently is crucial for many fields. In computing, algorithms often need to compare values quickly, and this method is a fast way to do it. It's used in data analysis to compare proportions and ratios, which is vital for careers in data science and even in designing smart AI systems that make decisions based on comparisons.

Common Mistakes

MISTAKE: Multiplying numerator by numerator and denominator by denominator. | CORRECTION: Always multiply the numerator of one fraction by the denominator of the OTHER fraction.

MISTAKE: Not associating the cross-product with the correct fraction. For example, when comparing a/b and c/d, putting (a*d) above c/d instead of a/b. | CORRECTION: The product (numerator of first * denominator of second) always belongs to the FIRST fraction. The product (numerator of second * denominator of first) always belongs to the SECOND fraction.

MISTAKE: Forgetting which sign to use after finding the cross-products. | CORRECTION: If the number above the first fraction is smaller, then the first fraction is smaller. If it's larger, the first fraction is larger.

Practice Questions

Try It Yourself

QUESTION: Which fraction is greater: 4/9 or 5/11? | ANSWER: 5/11 is greater (4*11 = 44, 5*9 = 45. Since 44 < 45, 4/9 < 5/11)

QUESTION: Arrange these fractions from smallest to largest: 1/2, 3/7, 2/5. | ANSWER: 3/7, 2/5, 1/2 (Compare 1/2 and 3/7: 7 vs 6, so 3/7 < 1/2. Compare 3/7 and 2/5: 15 vs 14, so 2/5 < 3/7 is wrong, 3/7 < 2/5 is wrong. 3/7 < 2/5 is wrong. (3*5=15, 2*7=14, so 2/5 < 3/7 is correct. My bad. Let's re-evaluate) (1/2 vs 3/7: 7 vs 6, so 3/7 < 1/2. 1/2 vs 2/5: 5 vs 4, so 2/5 < 1/2. Now 3/7 vs 2/5: 15 vs 14, so 2/5 < 3/7. Thus, 2/5 < 3/7 < 1/2.)

QUESTION: Your mobile data plan gives you 7/8 GB for the week. Your friend gets 5/6 GB. Who has more data? | ANSWER: You have more data (7*6 = 42, 5*8 = 40. Since 42 > 40, 7/8 > 5/6)

MCQ

Quick Quiz

Which of the following statements is true when comparing 3/4 and 5/7 using cross-multiplication?

3*7 = 21, 5*4 = 20, so 3/4 < 5/7

3*7 = 21, 5*4 = 20, so 3/4 > 5/7

3*4 = 12, 5*7 = 35, so 3/4 > 5/7

3*7 = 21, 5*4 = 20, so 21 = 20

The Correct Answer Is:

When you cross-multiply, 3*7 = 21 (for 3/4) and 5*4 = 20 (for 5/7). Since 21 is greater than 20, it means 3/4 is greater than 5/7.

Real World Connection

In the Real World

In cricket analytics, statisticians often compare batting strike rates or bowling averages, which are essentially fractions. For example, comparing a batsman who scored 45 runs in 30 balls (45/30) with another who scored 60 runs in 40 balls (60/40) can be done quickly using cross-multiplication to see who has a better strike rate. This helps coaches make decisions during matches.

Key Vocabulary

Key Terms

FRACTION: A part of a whole, written as a numerator over a denominator. | NUMERATOR: The top number in a fraction, showing how many parts are being considered. | DENOMINATOR: The bottom number in a fraction, showing the total number of equal parts. | CROSS-MULTIPLICATION: A method of multiplying the numerator of one fraction by the denominator of another. | COMPARE: To examine two or more items to establish their similarities or differences.

What's Next

What to Learn Next

Now that you can compare fractions, you're ready to learn about adding and subtracting fractions with different denominators. This skill builds directly on comparing, as finding common denominators for addition/subtraction uses similar logic to understanding fraction sizes.