Question

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

The ratio of $a$ to $c$ is $\frac{4}{15}$.

Step-by-step solution

- 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 $\frac{b}{c}=\frac{1}{5}$.

- 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 $c$.

- Substitute $b$ in the first ratio

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

- Simplify the ratio

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

- Find the ratio of $a$ to $c$

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

Key concepts

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.