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Chapter 6: Problem 417

In the following exercises, translate to a proportion. 42 is 70\(\%\) of what number?

Short Answer

Expert verified
60

Step by step solution

01

Understand the Problem

The problem is asking for a number which 42 is 70% of. To find this number, a proportion can be set up where 42 is some part of a whole number.
02

Set Up the Proportion

Write the proportion for the given percentage. In this case, the proportion will be set up as follows: \[ \frac{70}{100} = \frac{42}{x} \]Here, 70 represents the percentage, 100 is the whole, 42 is the part, and \(x\) is the unknown whole number.
03

Solve for x

To solve for \(x\), cross-multiply and solve the resulting equation. \[ 70x = 42 \times 100 \]This simplifies to: \[ 70x = 4200 \]Now, divide both sides by 70: \[ x = \frac{4200}{70} = 60 \]

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Key Concepts

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

Proportion
A proportion is an equation that states that two ratios are equal. In percentage problems like the one given, we often use proportions to relate the part with the whole through the percentage. To set up a proportion, you need to compare two ratios. For example, in this problem, we want to find out what number is the whole when 42 is 70% of it. We set it up as follows:
  • The percentage (70%) can be written as a fraction: \(\frac{70}{100}\).
  • The known value (42) is part of the unknown whole (x), so we write it as: \(\frac{42}{x}\).
Putting these ratios together, our proportion looks like this: \(\frac{70}{100} = \frac{42}{x}\). This equation shows that the relationship between 70 and 100 is the same as the relationship between 42 and the unknown number x.
Cross-Multiplication
Cross-multiplication is a method used to solve proportions. When you have a proportion like \(\frac{a}{b} = \frac{c}{d}\), you can solve for one of the variables by cross-multiplying. This means you multiply the numerator of one fraction by the denominator of the other fraction. Let's apply this to our example:
  • We have the proportion: \(\frac{70}{100} = \frac{42}{x}\).
  • To solve it, we cross-multiply to get: 70x = 42 \cdot 100\.
  • This simplifies to: 70x = 4200.
Now, you have a simple equation that you can solve for x. Cross-multiplication is a powerful tool for solving proportional relationships quickly and effectively.
Solving for Unknowns
Once you have set up the proportion and used cross-multiplication, you need to solve for the unknown variable. In this problem, we need to find 'x', which represents the whole number we are looking for. Here鈥檚 how you do it:
  • From the cross-multiplication step, you have 70x = 4200.
  • To find x, divide both sides of the equation by 70: \(\frac{4200}{70}\).
  • This simplifies to: x = 60.
So, 42 is 70% of 60. Finding the unknown in a proportion often involves isolating the variable on one side of the equation, which can be accomplished through basic algebraic techniques like division or multiplication. Solving for unknowns is a fundamental skill in algebra, especially when working with proportions and percentages.

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