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Chapter 6: Problem 45

Solve the proportion. Be sure to check your answers. $$\frac{3}{40}=\frac{w}{10}$$

Short Answer

Expert verified
w = 0.75

Step by step solution

01

Cross Multiply

To solve the proportion \(\frac{3}{40}=\frac{w}{10}\), begin by cross-multiplying the two fractions. This means you will multiply the numerator of the first fraction by the denominator of the second fraction and the numerator of the second fraction by the denominator of the first fraction. Thus, you get: \ 3 \times 10 = 40 \times w. \ Which simplifies to 30 = 40w.
02

Solve for w

Next, solve for \( w \) by isolating it on one side of the equation. To do this, divide both sides of the equation by 40. Hence, \ w = \frac{30}{40} \ simplifies to \ w = \frac{3}{4} \ or \ w = 0.75.
03

Check the Solution

To verify our solution, substitute \( w = 0.75 \) back into the original proportion: \( \frac{3}{40} = \frac{0.75}{10} \). Simplifying \( \frac{0.75}{10} \) gives us \( \frac{3}{40} \). Both fractions are equal, so the solution, \ w = 0.75 \, is correct.

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Key Concepts

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

cross-multiplication
Cross-multiplication is a technique used to solve proportions. A proportion is an equation stating that two ratios are equal. In our exercise, we are given the proportion \( \frac{3}{40} = \frac{w}{10} \). To solve it, we use cross-multiplication.

Here鈥檚 how it works: multiply the numerator of the first fraction by the denominator of the second fraction. Then, multiply the numerator of the second fraction by the denominator of the first fraction. For our given problem, this is executed as follows:
\[ 3 \times 10 = 40 \times w \]
This step simplifies our proportion into a simple equation:
\[ 30 = 40w \]
Now, we can proceed to solving this simpler equation.
solving for variables
Once we have an equation from our cross-multiplication step, we can then solve for the unknown variable. In our example, we have the equation \( 30 = 40w \). The goal is to isolate the variable \( w \) on one side of the equation.
To do this, we divide both sides of the equation by 40:
\[ w = \frac{30}{40} \]
This fraction can be simplified. Dividing both the numerator and the denominator by their greatest common divisor (10) gives us:
\[ w = \frac{3}{4} \]
This means \( w \) is equal to 0.75.
This technique鈥攕olving for variables鈥攃an be applied to any similar problem.
checking solutions
After finding the solution for \( w \), it is crucial to check that it is correct. This verification step ensures you haven't made any mistakes.
Substitute your found value back into the original proportion to verify it satisfies the initial equation. For our example, substitute \( w = 0.75 \) into the initial proportion \(\frac{3}{40} = \frac{w}{10} \):
\[ \frac{3}{40} = \frac{0.75}{10} \]
Simplify \( \frac{0.75}{10} \) to be sure:
\[ \frac{0.75}{10} = \frac{3}{40} \]
Since both sides of the proportion are equal, our solution \( w = 0.75 \) is verified as correct.
Thus, always check your solutions to confirm they work in the original equation.

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Most popular questions from this chapter

Determine whether the given value is a solution to the proportion. $$\frac{x}{40}=\frac{1}{-8} ; \quad x=-5$$

Yellowstone National Park in Wyoming has the largest population of free- roaming bison. To approximate the number of bison, 200 are captured and tagged and then let free to roam. Later, a sample of 120 bison is observed and 6 have tags. Approximate the population of bison in the park.

a. The price per share of Microsoft stock rose \(\$ 18.24\) in a 24 -month period. Compute the rate representing the increase in the price per month. b. The price per share of IBM stock rose \(\$ 22.80\) in a 12 -month period. Compute the rate representing the increase in the price per month. c. Which stock had a greater rate of increase per month?

Use \(=\) or \(\neq\) to make a true statement. $$\frac{3}{-13} \square \frac{-15}{65}$$

Determine whether the ratios form a proportion. $$\frac{5}{18}=\frac{4}{16}$$
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