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If the ratio of \(a\) to \(b\) is 4 to 3 and the ratio of \(b\) to \(c\) is 1 to \(5,\) what is the ratio of \(a\) to \(c ?\) $$\begin{array}{l} \frac{4}{5} \\ \frac{1}{3} \\ \frac{2}{5} \\ \frac{4}{5} \\ \frac{7}{6} \end{array}$$

Short Answer

Expert verified
The ratio of \(a\) to \(c\) is \(\frac{4}{15}\).

Step by step solution

01

- Identify given ratios

The ratio of \(a\) to \(b\) is given as \(4:3\), which can be written as \(\frac{a}{b} = \frac{4}{3}\). The ratio of \(b\) to \(c\) is given as \(1:5\), which can be written as \(\frac{b}{c} = \frac{1}{5}\).
02

- Express one ratio in terms of a common variable

From \(\frac{b}{c} = \frac{1}{5}\), we have \(b = \frac{c}{5}\). We can substitute this into the first ratio to express \(a\) in terms of \(c\).
03

- Substitute \(b\) in the first ratio

Substitute \(b = \frac{c}{5}\) into \(\frac{a}{b} = \frac{4}{3}\), giving us \(\frac{a}{\frac{c}{5}} = \frac{4}{3}\).
04

- Simplify the ratio

Simplify \(\frac{a}{\frac{c}{5}} = \frac{4}{3}\) by multiplying both sides by \(\frac{c}{5}\). This gives \(a = \frac{4}{3} \cdot \frac{c}{5} = \frac{4c}{15}\).
05

- Find the ratio of \(a\) to \(c\)

Since \(a = \frac{4c}{15}\), the ratio of \(a\) to \(c\) is \(\frac{a}{c} = \frac{\frac{4c}{15}}{c} = \frac{4}{15}\).

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Key Concepts

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

Ratio and Proportion
Understanding the concept of ratio and proportion is crucial to solving problems like these. In this exercise, we have two given ratios: the ratio of \(a\) to \(b\) is 4 to 3, and the ratio of \(b\) to \(c\) is 1 to 5. Ratios compare quantities relative to each other. To solve it, we express one of the ratios using a common variable.
We start by noting that the ratio of \(a\) to \(b\) can be written as \(\frac{a}{b} = \frac{4}{3}\). This means for every 4 units of \(a\), there are 3 units of \(b\). Similarly, for \(b\) to \(c\), \(\frac{b}{c} = \frac{1}{5}\), showing for every 1 unit of \(b\), there are 5 units of \(c\). By finding a common variable and substituting effectively, we can simplify the problem.
Algebraic Manipulation
Algebraic manipulation is managing and rearranging equations to isolate variables and simplify expressions.
From the second ratio \(\frac{b}{c} = \frac{1}{5}\), we derive \(b = \frac{c}{5}\). To find the ratio of \(a\) to \(c\), we substitute \(b = \frac{c}{5}\) into the first ratio. Hence, \(\frac{a}{b} = \frac{4}{3}\) becomes \(\frac{a}{\frac{c}{5}} = \frac{4}{3}\).
Next, we simplify by multiplying both sides by \(\frac{c}{5}\), resulting in \(a = \frac{4}{3} \cdot \frac{c}{5} = \frac{4c}{15}\). Consequently, the ratio of \(a\) to \(c\) is \(\frac{4}{15}\). By carefully applying these steps, the problem is methodically broken down, making the final solution precise and graspable.
Test Preparation
Proper test preparation is key to mastering problems like these. To excel, practice solving different types of ratio problems and familiarize yourself with algebraic manipulations.
Begin by reviewing core concepts regularly and employ mnemonic strategies to remember key formulas. Practice identifying and substituting common variables. Understand the process of transitioning from one ratio to another by expressing them in terms of a common variable. This will help in solving multi-step problems efficiently.
Utilize daily practice problems, mock GMAT exams, and reviewing detailed solutions to reinforce learning. Lastly, time yourself when solving these problems to improve speed and accuracy under exam conditions.

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Most popular questions from this chapter

Three investors, \(A, B,\) and \(C,\) divide the profits from a business enterprise in the ratio of \(5: 7: 8,\) respectively. If investor A earned \(\$ 3,500,\) how much money did investors \(\mathrm{B}\) and \(\mathrm{C}\) earn in total? $$\begin{array}{c} \$ 4,000 \\ \$ 4,900 \\ \$ 5,600 \\ \$ 9,500 \\ \$ 10,500 \end{array}$$

In \(2010,\) a basketball team won 30 percent of its 20 basketball games. In 2011 , the team won 28 percent of its 25 basketball games. What was the percent increase from 2010 to 2011 in the number of basketball games the team won? $$\begin{array}{l} 5 \% \\ 14 \frac{2}{7} \% \\ 16 \frac{2}{3} \% \\ 23 \frac{1}{3} \% \\ 60 \% \end{array}$$

The cost of Brand V paper is proportional to the weight. If 18 ounces of Brand \(V\) paper cost \(\$ 3.06,\) what is the cost of 24 ounces of Brand V paper? $$\begin{array}{l} \$ 4.08 \\ \$ 5.58 \\ \$ 6.12 \\ \$ 7.31 \\ \$ 7.59 \end{array}$$

How many liters of water must be evaporated from 50 liters of a 3 percent sugar solution to get a 5 percent sugar solution? $$\begin{array}{l} 2 \\ 4 \\ 6 \\ 10 \\ 20 \end{array}$$

Each employec of Company \(\mathrm{X}\) is a member of precisely 1 of 3 shifts of employees. Among the 60 members of the first shift, 20 percent participate in the pension program; among the 50 members of the second shift, 40 percent participate in the pension program; and among the 40 members of the third shift, 10 percent participate in the pension program. What percent of the workers at Company X participate in the pension program? $$\begin{array}{l} 20 \% \\ 24 \% \\ 36 \% \\ 37.5 \% \\ 70 \% \end{array}$$

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