Comparing Quantities

Introduction

In our daily lives, we often compare different things. Whether it is comparing prices between different products, checking discounts during a sale, or comparing sports statistics, the concept of comparing quantities is all around us. This lesson will focus on understanding how to compare quantities using numbers, ratios, percentages, and more. By the end of this lesson, you will be comfortable working with these concepts in mathematics and applying them to real-world scenarios.

Basic concepts

Understanding ratios

A ratio is a comparison between two quantities. It tells us how much of one thing is compared to another. Ratios are used in many different aspects of life, such as recipe making, map reading, and financial analysis.

A basket contains apples and oranges in the ratio 3 : 2.

This means that for every 3 apples there are 2 oranges.

To express this in mathematical terms, the ratio of 3:2 can be written as:

3/2

Ratio

A ratio is an equation that states that two ratios are equal. For example, if there are 3 apples for every 2 oranges, then there should be 6 apples for every 4 oranges if the ratio is maintained.

3/2 = 6/4

Percentage

Percentages are another way to compare quantities, specifically in terms of 100. It expresses a number as a fraction of 100 using the % symbol.

For example, to find out what percent 25 is of 200:

(25/200) * 100 = 12.5%

This means 25 is 12.5% of 200.

Application of Percentages

Increase and decrease

Percentages are often used to calculate the increase or decrease in given quantities, such as prices or population.

Extend the instance

If the price of a shirt increases from $50 to $60, we would calculate the percentage increase as follows:

Increase = New Price - Old Price = $60 - $50 = $10
Percentage increase = (Increase/Old price) * 100 = (10/50) * 100 = 20%

Subtract example

If the price of a book decreases from $30 to $24, we calculate the percentage decrease:

Shortage = Old price - New price = $30 - $24 = $6
Percentage reduction = (Reduction/Old price) * 100 = (6/30) * 100 = 20%

Interest calculation

In financial terms, percentages play an important role in determining interest on loans and investments. The two main types of interest are simple interest and compound interest.

Simple interest

Simple interest is calculated directly on the principal amount. Its formula is:

Simple Interest (SI) = (Principal (P) * Rate (r) * Time (t)) / 100

For example, if you invest $1,000 at a 5% annual interest rate for 3 years, the interest earned would be:

SI = (1000 * 5 * 3) / 100 = $150

Compound interest

Compound interest is calculated on the initial principal and the accumulated interest of previous periods. Its formula is:

Compound Interest (CI) = P(1 + R/100)^T - P

If you invest $1,000 at a 5% annual interest rate for 3 years, the compound interest will be:

CI = 1000(1 + 5/100)^3 - 1000 = $157.63

Real-life applications

It is very important to understand how to compare quantities in different areas. Let's take a look at some real-life examples where this knowledge comes in handy.

Discount on purchases

When shopping, you often encounter discounts. Being able to calculate discounts and compare prices helps in making decisions. For example, if a pair of shoes is labeled 25% off and its original price is $120:

Discount amount = Original price * (Discount percentage / 100) = 120 * (25/100) = $30
Price after discount = Original price - Discount amount = 120 - 30 = $90

Cooking recipes

When following a cooking recipe, you may need to adjust the quantities based on the number of servings you want. Ratios are important for measuring recipes. If the original recipe is for 4 servings, but you need 6:

Ratio = New servings / Original servings = 6/4 = 1.5
Multiply each factor by 1.5

Map reading

Maps often use a scale expressed as a ratio, such as 1:100,000, which means that 1 cm on the map is equivalent to 100,000 cm in real life. If the distance on the map is 5 cm:

Actual distance = 5 * 100,000 = 500,000 cm = 5 km

Common mistakes to avoid

People often make some common mistakes while comparing quantities. Here are ways to avoid them:

• Make sure you have consistent units when comparing quantities. Convert units when necessary.
• Double-check calculations, especially percentages and interest, as misplaced decimals can introduce errors.
• Clarify whether you are dealing with gross or net figures, especially in a financial context.
• Misinterpreting the order in a ratio. A 3:2 ratio of apples and oranges means 3 apples compared to 2 oranges, not the other way round.

Practice exercises

Practice is vital to master the concept of comparing quantities. Try solving the following problems:

Exercise 1: Ratio

There are 15 boys and 20 girls in a class. Express the ratio of boys and girls and the ratio of girls to the total number of students.

Solution:
Ratio of boys and girls = 15:20 = 3:4 
Ratio of girls to total students = 20:(15+20) = 20:35 = 4:7

Exercise 2: Percentages

A smartphone is being sold at a 30% discount. If the original price was $850, what is the new price?

Solution:
Discount = 850 * (30/100) = $255
New price = 850 - 255 = $595

Exercise 3: Simple and Compound Interest

Calculate the simple and compound interest on a sum of $2,000 for 5 years at an annual interest rate of 4%.

Solution:
Simple Interest (SI) = (2000 * 4 * 5) / 100 = $400

Compound Interest (CI) = 2000(1 + 4/100)^5 - 2000 = $433.05

Conclusion

Comparing quantities is a basic skill in math that applies to almost every aspect of life. By understanding the concepts of ratios, percentages, interest rates, and more, you can make informed decisions in practical situations, whether it's at the supermarket, in finance, or while navigating a route. With consistent practice, the process of comparing quantities will become intuitive and an indispensable part of your problem-solving toolkit.

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