Finding the Ratio of Numbers with Percentage Differences

Let's break down this problem about finding the ratio between two numbers when they are compared to a third number using percentages.

We are given that two numbers are, respectively, 10% and 25% more than a third number. We need to find the ratio of these two numbers.

Step 1: Define the Third Number

Let the third number be represented by a variable. A common variable for an unknown value is \(x\).

Third Number = \(x\)

Step 2: Calculate the First Number

The first number is 10% more than the third number. This means we add 10% of the third number to the third number itself.

10% of \(x\) can be written as \(\frac{10}{100} \times x = 0.10x\).

So, the first number is \(x + 0.10x\).

First Number = \(x(1 + 0.10) = 1.10x\)

Step 3: Calculate the Second Number

The second number is 25% more than the third number. This means we add 25% of the third number to the third number.

25% of \(x\) can be written as \(\frac{25}{100} \times x = 0.25x\).

So, the second number is \(x + 0.25x\).

Second Number = \(x(1 + 0.25) = 1.25x\)

Step 4: Find the Ratio of the Two Numbers

We need to find the ratio of the first number to the second number. A ratio can be expressed as a fraction.

Ratio = \(\frac{\text{First Number}}{\text{Second Number}}\)

Substitute the values we found for the first and second numbers:

Ratio = \(\frac{1.10x}{1.25x}\)

Step 5: Simplify the Ratio

The variable \(x\) appears in both the numerator and the denominator, so they cancel each other out.

Ratio = \(\frac{1.10}{1.25}\)

To get rid of the decimals and simplify the ratio, we can multiply both the numerator and the denominator by 100.

Ratio = \(\frac{1.10 \times 100}{1.25 \times 100} = \frac{110}{125}\)

Now, we need to simplify the fraction \(\frac{110}{125}\) by finding the greatest common divisor (GCD) of 110 and 125. Both numbers are divisible by 5.

• Divide the numerator by 5: \(110 \div 5 = 22\)
• Divide the denominator by 5: \(125 \div 5 = 25\)

The simplified fraction is \(\frac{22}{25}\).

The ratio of the two numbers is 22 : 25.

Summary of Calculations

The ratio of the two numbers is 22 : 25.

Revision Table: Percentage and Ratio Concepts

Additional Information: Working with Percentages and Ratios

Understanding percentages and ratios is fundamental in quantitative problems. A percentage is essentially a fraction out of 100. For example, 10% is \(\frac{10}{100}\) or 0.10.

When a number is 'X% more than' another number, it means the original number is increased by X% of itself. If the original number is \(N\), the new number is \(N + \frac{X}{100}N = N(1 + \frac{X}{100})\). This shortcut \(N(1 + \frac{X}{100})\) is very useful for quick calculations of percentage increases.

Ratios allow us to compare quantities proportionally. The ratio 22:25 means that for every 22 units of the first number, there are 25 units of the second number. This relationship holds true regardless of the actual value of the third number \(x\) (as long as \(x\) is not zero).

In this problem, we used a variable \(x\) for the third number. We could also have assumed the third number to be 100 for simplicity, as ratios are independent of the absolute values as long as the proportions are maintained.

• If third number = 100
• First number = 100 + 10% of 100 = 100 + 10 = 110
• Second number = 100 + 25% of 100 = 100 + 25 = 125
• Ratio = \(\frac{110}{125} = \frac{22}{25}\) or 22:25

Both methods (using \(x\) or assuming 100) yield the same ratio, confirming the result.