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Chapter 2: Problem 32

Solve each equation. $$ \frac{20}{100}=\frac{z}{80} $$

Short Answer

Expert verified
z = 16.

Step by step solution

01

- Understand the Proportion

The equation is given in the form of a proportion: \( \frac{20}{100} = \frac{z}{80} \). This means that the ratio of 20 to 100 is the same as the ratio of \( z \) to 80.
02

- Cross Multiply

Cross-multiplication is a common method to solve proportions. Multiply the numerator of each fraction by the denominator of the other fraction: \( 20 \times 80 = 100 \times z \).
03

- Solve for z

Simplify the equation: \( 1600 = 100z \). To find the value of \( z \), divide both sides by 100: \( z = \frac{1600}{100} \).
04

- Simplify the Result

Simplify the fraction to get the final value of \( z \): \( z = 16 \).

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Key Concepts

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

Cross Multiplication
Imagine you have two fractions set equal to each other, like this: \[ \frac{20}{100} = \frac{z}{80} \] In this type of proportion, you can apply cross multiplication to find the missing value. Cross multiplication means you multiply the numerator of one fraction by the denominator of the other fraction. Here's how it works:
\[ 20 \times 80 = 100 \times z \] This gives you a simpler equation to solve. Simplifying further, you get \[ 1600 = 100z \]. By isolating the variable z, you can easily find its value.
Ratio and Proportion
A ratio compares two quantities, showing the relative size of one quantity to another. In our problem, we have the ratio 20:100. Using ratios helps us work with proportions easily. A proportion states that two ratios are equal.
For example, \[ \frac{20}{100} \text{ is equal to } \frac{z}{80} \] indicates that the ratio of 20 to 100 is the same as the ratio of z to 80. Setting these ratios in equation form allows us to find the unknown variable using the cross multiplication method. Proportions help us understand relationships between numbers in a clear and simple way.
Simplifying Fractions
After finding the value of z using cross multiplication, we must simplify the fraction to get the final answer. Simplifying means reducing the fraction to its simplest form. Once we have: \[ z = \frac{1600}{100} \], we divide both the numerator and the denominator by their greatest common divisor. In this case, we divide 1600 and 100 by 100, giving us: \[ z = \frac{1600 \text{ divided by } 100}{100 \text{ divided by } 100} = 16 \]. Simplifying fractions makes it easier to interpret the values at hand.

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