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Chapter 2: Problem 96

Solve each problem. \(140 \%\) of what number is \(315 ?\)

Short Answer

Expert verified
225

Step by step solution

01

Understand the problem

The problem asks for a number such that 140% of this unknown number equals 315.
02

Set up the equation

Let the unknown number be denoted by \( x \). The equation based on the problem statement is: 0. Use the percent notation to decimal conversion: 140% = 1.4 Thus, the equation becomes: 1.4x = 315
03

Solve for the unknown number

To find \( x \), divide both sides of the equation by 1.4: \( x = \frac{315}{1.4} \) After performing the division, we get: \( x = 225 \)
04

Verify the solution

To ensure the solution is correct, check if 140% of 225 equals 315: 1.4 * 225 = 315 The calculated value matches the given condition, so the solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

percent to decimal conversion
When solving percentage problems, converting a percent to a decimal is often the first step. Percent means 'per hundred,' so we convert by moving the decimal point two places to the left.
For example, to convert 140% to a decimal, follow these steps:
  • Write 140 as a fraction, which is 140/100.
  • Simplify the fraction to 1.4.
This means 140% is 1.4 in decimal notation.
Understanding this allows you to easily manage percentages in equations and other math problems.
equation setup
Setting up the equation correctly based on the problem statement is crucial. This ensures you solve the right problem. Let's look at our example:
The problem states: 140% of what number is 315?
We denote the unknown number as x.
Then, we set up the equation as 1.4x = 315, where 1.4 is the decimal conversion of 140%.
This kind of setup helps you translate a word problem into a mathematical equation that is easier to solve.
solving linear equations
After setting up your equation, the next step is solving it. Linear equations involve finding the value of the unknown by isolating it on one side of the equation. From our setup:
1.4x = 315
To isolate x, you need to divide both sides by 1.4:
\[ x = \frac{315}{1.4} \]
Calculating the right side, we get:
\[ x = 225 \]
Thus, the unknown number that when multiplied by 140% gives 315 is 225. Breaking it into steps makes solving more manageable.
verification of solutions
Verifying your solution ensures that your math is correct. Let's check our example:
We found that x = 225.
We need to confirm if 140% of 225 equals 315. First, convert 140% to a decimal (1.4) and then multiply by 225:
\[ 1.4 \times 225 = 315 \]
The calculated result matches the given number. So, our solution is correct!
Verification is an essential step to confirm your answer and reinforce your understanding.

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