Problem 83 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the given information to write an equation. Let \(x\) represent the number described in each exercise. Then solve the equation and find the number. When one-third of a number is added to one-fifth of the number, the sum is \(16 .\) What is the number?

Step-by-Step Solution

Answer

The number is 30.

Step 1: Define variable

Let \(x\) be the number we are trying to find.

Step 2: Formulate equation

According to the problem, when one-third of this number is added to one-fifth of the number, the result is 16. This information can be formulated into the following equation: \(\frac{1}{3}x + \frac{1}{5}x = 16\).

Step 3: Solve the equation

First add the fractions on the left side of the equation. Both fractions have the same variable, so the fractions can be added directly. The result of the addition is \(\frac{8}{15}x = 16\). Next, to isolate \(x\), divide both sides of the equation by the fraction \(\frac{8}{15}\). The result is \(x = 16 \div \frac{8}{15}\). Make the division easier by multiply the numerator and denominator by 15. So, \(x = 16 \times \frac{15}{8} = 30\).

Step 4: Check the solution

Substitute \(x = 30\) into the original equation, which gives \(\frac{1}{3} \times 30 + \frac{1}{5} \times 30 = 16\). The check proves to be valid.

Key Concepts

Algebra

Algebra is all about finding the unknown or putting real life variables into equations and then solving them. In this exercise, we are tasked to find a number that fits a certain described condition.
We do this by setting up an equation. The equation is a mathematical statement that uses an equal sign: what is on the left side must be equal to the right side. With algebra, we can manipulate these expressions to find the unknown number.
Here, we started by defining the number we need to find as a variable, which is the norm in algebra. In this example, we called it \(x\). The problem states that one-third of this number added to one-fifth of this number equals 16. This setup forms our starting equation.

Fractions

Fractions represent a part of a whole and are fundamental in algebra for solving various equations. In our exercise, fractions show up because we are dealing with parts of a number. Specifically, one-third \(\left(\frac{1}{3}\right)\) and one-fifth \(\left(\frac{1}{5}\right)\) of our unknown number \(x\).
To solve the equation \(\frac{1}{3}x + \frac{1}{5}x = 16\), we first need to have a common denominator for the fractions. This allows us to add them together easily.

• The least common denominator of 3 and 5 is 15, allowing us to rewrite and add these fractions.
• Multiply each fraction by what would make it into \(\frac{15}{15}\). \(\frac{1}{3} = \frac{5}{15}\) and \(\frac{1}{5} = \frac{3}{15}\).
• Once rewritten, you add the fractions: \(\frac{5}{15}x + \frac{3}{15}x = \frac{8}{15}x\).

Next, you need to solve for \(x\) which involves dividing 16 by \(\frac{8}{15}\). Perform this calculation by multiplying instead of dividing, using the reciprocal of \(\frac{8}{15}\). Thus, \(x = 16 \times \frac{15}{8} = 30\).

Variable Representation

Variables play a crucial role in algebra as they stand in for unknown values. They allow us to form equations based on real-world situations, even when not all values are initially known. In this exercise, \(x\) represents our unknown number.

• Defining the variable helps to structure the equation. Here, we set \(x\) as the number of interest.
• By understanding this representation, we can focus on solving the equation step by step, concentrating on isolating \(x\).
• This structured approach simplifies complex word problems, making them more manageable to solve step by step.

Another important aspect of using variables is checking the solution, which helps us verify that our calculated \(x\) satisfies the given conditions of the problem, eventually confirming the solution is correct.