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Chapter 6: Problem 59

Write a proportion for each statement. Then solve for the variable. 25 is to 100 as 9 is to \(y\)

Short Answer

Expert verified
y = 36.

Step by step solution

01

Set Up the Proportion

First, write the given ratio as a proportion. The statement '25 is to 100 as 9 is to y' can be set up as \[\frac{25}{100} = \frac{9}{y}\].
02

Cross Multiply

To solve for the variable, cross multiply the fractions. This means multiplying the numerator of one fraction by the denominator of the other fraction:\[25 \times y = 100 \times 9\]
03

Solve for y

To isolate the variable y, divide both sides of the equation by 25:\[y = \frac{100 \times 9}{25}\].
04

Simplify the Equation

Simplify the right-hand side of the equation to find y:\[y = \frac{900}{25} = 36\]
05

Conclusion

Therefore, the value of y that makes the proportion true is y = 36.

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Key Concepts

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

Ratios
In math, a ratio compares two quantities to show which one is larger or smaller and by how much. For example, the ratio of 25 to 100 can be written as \(\frac{25}{100}\) or 25:100. Ratios help us compare different things, but they need to be in the same units. Here, 25 and 100 are both numbers, so we can directly compare them. To set up a proportion, we use two ratios and set them equal to each other. This forms an equation that we can solve to find an unknown value.
Cross Multiplication
Cross multiplication is a simple method to solve proportions. When we have a proportion like \(\frac{25}{100} = \frac{9}{y}\), we can cross-multiply to find the value of y. This means we multiply the numerator (top number) of the first fraction by the denominator (bottom number) of the second fraction and vice versa. Here, we get the equation \[25 \times y = 100 \times 9 \]. This step eliminates the fractions and gives us a straightforward equation to solve.
Solving for Variables
After cross-multiplying, we need to isolate the variable y. Isolating means getting the variable by itself on one side of the equation. In our example \[25 \times y = 100 \times 9 \], we already cross-multiplied. Now, divide both sides of the equation by 25 to isolate y: \[ y = \frac{100 \times 9}{25} \]. This solves for y by performing the division. Here, the calculation simplifies to \[ y = 36 \].
Simplifying Fractions
Simplifying fractions makes them easier to work with. To simplify, divide the numerator and the denominator by their greatest common divisor (GCD). In this case, after we solve the equation \[ y = \frac{100 \times 9}{25} = \frac{900}{25} \], you'll notice the fraction \(\frac{900}{25}\). Simplify it by dividing the numerator and the denominator by their GCD, which is 25: \[ \frac{900}{25} = 36 \]. This means the simplified value of y is 36. Simplifying helps to check or confirm our answer.

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Most popular questions from this chapter

Refer to the table showing Alex Rodriguez's salary (rounded to the nearest \(\$ 100,000\) ) for selected years during his career. Write each ratio in lowest terms. \begin{array}{l|l|l|l} \hline \text { Year } & \text { Team } & \text { Salary } & \text { Position } \\ \hline 2007 & \text { New York Yankees } & \$ 22,700,000 & \text { Third baseman } \\ \hline 2004 & \text { New York Yankees } & \$ 22,000,000 & \text { Third baseman } \\ \hline 2000 & \text { Seattle Mariners } & \$ 4,400,000 & \text { Shortstop } \\ \hline 1996 & \text { Seattle Mariners } & \$ 400,000 & \text { Shortstop } \\ \hline \end{array} Write a ratio of the increase in Alex's salary between the years 1996 and the year 2000 to his salary in 1996 .

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Solve the proportion. $$\frac{-3 \frac{1}{2}}{k}=\frac{2 \frac{1}{3}}{4}$$

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