What is Solving Proportions using Cross-Multiplication?

S3-SA1-0621

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Solving proportions using cross-multiplication is a method to find an unknown value when two ratios are equal. It involves multiplying the numerator of one ratio by the denominator of the other ratio, and setting these products equal to each other.

Simple Example

Quick Example

Imagine you are making chai. If 2 cups of milk are needed for 4 cups of chai, how many cups of milk do you need for 8 cups of chai? Here, the ratio of milk to chai should stay the same. Cross-multiplication helps us find the unknown amount of milk.

Worked Example

Step-by-Step

QUESTION: If 3 packets of biscuits cost Rs 60, how much will 5 packets of biscuits cost?

STEP 1: Set up the proportion. Let 'x' be the unknown cost.
3 packets / Rs 60 = 5 packets / Rs x

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STEP 2: Cross-multiply the numbers.
3 * x = 60 * 5

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STEP 3: Simplify the multiplication.
3x = 300

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STEP 4: Isolate 'x' by dividing both sides by 3.
x = 300 / 3

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STEP 5: Calculate the value of x.
x = 100

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ANSWER: 5 packets of biscuits will cost Rs 100.

Why It Matters

Understanding proportions and cross-multiplication is key in many advanced fields. From designing efficient algorithms in Computer Science to calculating forces in Physics, or even predicting market trends in Economics, this skill is fundamental. It helps engineers build bridges and data scientists analyze huge datasets.

Common Mistakes

MISTAKE: Multiplying numerators together and denominators together instead of cross-multiplying. | CORRECTION: Always multiply the numerator of one fraction by the denominator of the other fraction across the equals sign.

MISTAKE: Not keeping the units consistent when setting up the proportion (e.g., rupees on top in one ratio, but packets on top in the other). | CORRECTION: Ensure that corresponding quantities (like 'packets' or 'cost') are in the same position (both numerators or both denominators) in both ratios.

MISTAKE: Forgetting to divide by the coefficient of 'x' at the end. | CORRECTION: After cross-multiplying and simplifying, remember to perform the final division to find the value of the unknown variable.

Practice Questions

Try It Yourself

QUESTION: If a car travels 120 km in 2 hours, how far will it travel in 3 hours at the same speed? | ANSWER: 180 km

QUESTION: A recipe for ladoos needs 2 cups of besan for 10 ladoos. How many cups of besan are needed for 25 ladoos? | ANSWER: 5 cups

QUESTION: A mobile data plan gives 1.5 GB for Rs 299 for 28 days. If you want a plan for 42 days with the same daily data rate, how much should it ideally cost? (Assume proportional cost). | ANSWER: Rs 448.50

MCQ

Quick Quiz

If 4 pencils cost Rs 20, what is the cost of 7 pencils?

Rs 25

Rs 30

Rs 35

Rs 40

The Correct Answer Is:

Set up the proportion: 4/20 = 7/x. Cross-multiply: 4x = 20 * 7, so 4x = 140. Divide by 4: x = 35.

Real World Connection

In the Real World

From calculating ingredient amounts for a family dinner to scaling up production in a factory, cross-multiplication is everywhere. A chef uses it to adjust recipes, a shopkeeper uses it to figure out bulk prices, and even apps like Zomato or Swiggy use proportional reasoning to estimate delivery times based on distance.

Key Vocabulary

Key Terms

PROPORTION: An equation stating that two ratios are equal. | RATIO: A comparison of two quantities by division. | CROSS-MULTIPLICATION: A method to solve proportions by multiplying diagonally across the equals sign. | UNKNOWN: The variable or value we are trying to find.

What's Next

What to Learn Next

Next, you can explore 'Direct and Inverse Proportion' to understand how quantities relate in different ways. This will build on your cross-multiplication skills and help you solve even more complex real-world problems.