Understanding the Percentage Increase Problem

The problem asks us to find the ratio of two numbers, each of which is a certain percentage more than a third number. We are given that the first number is 17% more than the third number, and the second number is 50% more than the third number.

Setting up the Numbers

Let's assume the third number is \(x\). This will help us express the other two numbers algebraically.

• The first number is 17% more than the third number.
• The second number is 50% more than the third number.

Calculating the Two Numbers

To find a number that is a certain percentage more than another, we add the percentage increase (as a decimal) to 1 and multiply it by the original number.

Let the third number be \(x\).

The first number is 17% more than \(x\). This means the first number is \(100\% + 17\% = 117\%\) of \(x\). In decimal form, \(117\%\) is \(1.17\). So, the first number is \(1.17 \times x\), or \(1.17x\).

The second number is 50% more than \(x\). This means the second number is \(100\% + 50\% = 150\%\) of \(x\). In decimal form, \(150\%\) is \(1.50\). So, the second number is \(1.50 \times x\), or \(1.50x\).

Finding the Ratio

We need to find the ratio of the two numbers. This is the ratio of the first number to the second number.

Ratio = First Number : Second Number

Substituting the expressions we found:

Ratio = \(1.17x : 1.50x\)

Simplifying the Ratio

To simplify the ratio \(1.17x : 1.50x\), we can divide both sides by \(x\) (assuming \(x\) is not zero, which it must be for percentage increase to make sense). Ratio = \(1.17 : 1.50\)

To get rid of the decimals, we can multiply both sides of the ratio by 100:

Ratio = \(1.17 \times 100 : 1.50 \times 100\)
Ratio = \(117 : 150\)

Now, we need to simplify the ratio \(117 : 150\) by finding the greatest common divisor (GCD) of 117 and 150 and dividing both numbers by it.

Let's find the prime factors of 117 and 150:

• \(117 = 3 \times 39 = 3 \times 3 \times 13\)
• \(150 = 2 \times 75 = 2 \times 3 \times 25 = 2 \times 3 \times 5 \times 5\)

The common prime factor is 3. So, the GCD of 117 and 150 is 3.

Now, divide both parts of the ratio by 3:

\(117 \div 3 = 39\)
\(150 \div 3 = 50\)

The simplified ratio is \(39 : 50\).

Step-by-Step Calculation Summary

1. Assume the third number is \(x\).
2. Calculate the first number: \(x \times (1 + 17/100) = x \times 1.17 = 1.17x\).
3. Calculate the second number: \(x \times (1 + 50/100) = x \times 1.50 = 1.50x\).
4. Form the ratio of the two numbers: \(1.17x : 1.50x\).
5. Simplify the ratio by dividing by \(x\): \(1.17 : 1.50\).
6. Multiply by 100 to remove decimals: \(117 : 150\).
7. Simplify by dividing by the GCD (3): \(117/3 : 150/3 = 39 : 50\).

Ratio Comparison

The ratio of the two numbers is \(39 : 50\).

Let's compare this with the given options to ensure accuracy.

Revision Table - Key Concepts

Additional Information - Percentage and Ratio Basics

Understanding Percentages: A percentage is a way of expressing a number as a fraction of 100. For example, 17% means 17 out of 100, or \(17/100 = 0.17\).

Calculating 'More Than': When a quantity is 'P% more than' another quantity, it means the new quantity is the original quantity plus P% of the original quantity. If the original quantity is Q, the new quantity is \(Q + (P/100) \times Q = Q(1 + P/100)\).

Ratio Properties: A ratio \(a:b\) can be scaled by multiplying or dividing both \(a\) and \(b\) by the same non-zero number \(k\). The ratio \(a:b\) is equivalent to \(ka:kb\).