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Chapter 6: Problem 303

In the following exercises, translate to a proportion. What number is \(150 \%\) of \(64 ?\)

Short Answer

Expert verified
The number is 96.

Step by step solution

01

Understand the Problem

You need to find out what number is 150% of 64. This can be translated into a mathematical equation where you are looking for a number that represents 150% of 64.
02

Set Up the Proportion

A percentage is a way of expressing a number as a fraction of 100. So, 150% can be written as \(\frac{150}{100}\). Set up the proportion as follows: \(\frac{150}{100} = \frac{x}{64}\), where \x\ represents the number you are looking for.
03

Solve the Proportion

Cross-multiply to solve for \(x\): \(150 \times 64 = 100 \times x\) \(9600 = 100x\).
04

Isolate \(x\)

To find \(x\), divide both sides of the equation by 100: \(x = \frac{9600}{100} = 96\).
05

Verify the Solution

To ensure the result is correct, check that 96 is indeed 150% of 64. Calculate \(1.5 \times 64\) which also gives 96.

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Key Concepts

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

Percentage Calculations
Understanding percentages is crucial in prealgebra. A percentage represents a part of a whole and is expressed as a fraction of 100. When we see something like 150%, it essentially means 150 out of 100. This can be compared to more basic fractions, where for instance, 50% would be written as \(\frac{50}{100}\) or simplified to \(\frac{1}{2}\).

In our scenario, you need to determine what number is 150% of 64. To convert 150% into a fraction, you write it as \(\frac{150}{100}\). This helps in setting up the equation to solve the problem.
Cross-Multiplication
Cross-multiplication is a handy method to solve proportions. A proportion is an equation where two ratios are set equal to each other. To set up our equation: \(\frac{150}{100} = \frac{x}{64}\), where \(x\) represents the unknown number you're solving for.

With cross-multiplying, you multiply the denominator of one fraction by the numerator of the other, and set these products equal. So, from our proportion, you get:
  • 150 (from the first fraction) times 64 (from the second fraction) = 9600
  • 100 (from the second fraction) times \(x\) (from the first fraction)
This step gives you the equation: \(150 \times 64 = 100 \times x\). Breaking down this step, you get: \(9600 = 100x\).
Solving Proportions
After setting up and cross-multiplying the proportion, the next step is isolating the variable to find its value. From the previous cross-multiplying step, you have: \(9600 = 100x\).

To solve for \(x\), divide both sides of the equation by 100. This gives: \(x = \frac{9600}{100}\), simplifying to: \(x = 96\). This means 96 is 150% of 64.

To ensure the answer is correct, it's good practice to verify by reverse calculation. Multiply 64 by 1.5 (which represents 150% as a decimal). You'll get the same result: \(1.5 \times 64 = 96\). This confirms our solution is correct.

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