Finding a Number Using Percentages and Sum

This question asks us to find a number based on information about percentages of that number. We are told that when we add 40% of the number and 30% of the same number, the result is 70. We need to figure out what the original number is.

Setting Up the Percentage Problem

Let the unknown number be represented by the variable \(x\). The problem gives us two parts:

• 40% of the number
• 30% of the number

The sum of these two parts is equal to 70. We can write percentages as decimals or fractions. 40% is equivalent to 0.40 or \(\frac{40}{100}\), and 30% is equivalent to 0.30 or \(\frac{30}{100}\).

So, 40% of \(x\) can be written as \(0.40x\) or \(\frac{40}{100}x\).

And 30% of \(x\) can be written as \(0.30x\) or \(\frac{30}{100}x\).

Formulating the Equation

The problem states that the sum of these two parts is 70. So, we can write the equation:

\(40\%\) of \(x + 30\%\) of \(x = 70\)

Using decimals, the equation is:

\(0.40x + 0.30x = 70\)

Solving the Equation to Find the Number

Now we need to solve this equation for \(x\).

Combine the terms on the left side:

\((0.40 + 0.30)x = 70\)

\(0.70x = 70\)

To find \(x\), divide both sides of the equation by 0.70:

\(x = \frac{70}{0.70}\)

To make the division easier, we can multiply the numerator and the denominator by 10 to remove the decimal:

\(x = \frac{70 \times 10}{0.70 \times 10}\)

\(x = \frac{700}{7}\)

Now, perform the division:

\(x = 100\)

So, the number is 100.

Verification of the Result

Let's check if our answer is correct by plugging \(x = 100\) back into the original statement:

40% of 100 is \(0.40 \times 100 = 40\).

30% of 100 is \(0.30 \times 100 = 30\).

The sum of 40% and 30% of 100 is \(40 + 30 = 70\).

This matches the information given in the question. Therefore, the number is indeed 100.

Summary of Steps

1. Identify the unknown number as \(x\).
2. Write the percentages as decimals or fractions.
3. Formulate the equation based on the given information.
4. Solve the equation for \(x\).
5. Verify the answer.

Comparing with Options

The options provided were:

1. 150
2. 125
3. 200
4. 100

Our calculated number is 100, which matches option 4.

Revision Table: Key Concepts

Additional Information: Understanding Percentages

Percentages are a fundamental concept in mathematics used widely in everyday life, from calculating discounts and taxes to understanding statistics and grades. A percentage is essentially a way of expressing a proportion out of 100.

• To convert a percentage to a decimal, divide by 100 (or move the decimal point two places to the left). Example: 50% = 50/100 = 0.50.
• To convert a decimal to a percentage, multiply by 100 (or move the decimal point two places to the right). Example: 0.75 = 0.75 * 100 = 75%.
• To find a percentage of a number, convert the percentage to a decimal or fraction and multiply by the number. Example: 25% of 200 is \(0.25 \times 200 = 50\) or \(\frac{25}{100} \times 200 = 50\).

In this problem, combining percentages of the same number simplifies the calculation. 40% of \(x\) plus 30% of \(x\) is simply \((40\% + 30\%)\) of \(x\), which is 70% of \(x\). So, \(70\% \text{ of } x = 70\), leading to the same equation \(0.70x = 70\).