Problem 50 - Prealgebra

Question

Simplify each expression as much as possible. If the quotient of 50 and \(\frac{5}{3}\) is increased by \(8,\) what number results?

Step-by-Step Solution

Answer

The number is 38.

Step 1: Understand the Problem

We have to simplify an expression where we first find the quotient of two numbers and then increase it by another number. The expression is the quotient of 50 and \( \frac{5}{3} \), increased by 8.

Step 2: Divide 50 by \( \frac{5}{3} \)

To divide by a fraction, multiply by its reciprocal. The reciprocal of \( \frac{5}{3} \) is \( \frac{3}{5} \). So, calculate \( 50 \times \frac{3}{5} \).

Step 3: Calculate the Multiplication

Compute \( 50 \times \frac{3}{5} \). This simplifies to \( \frac{150}{5} \), which equals 30.

Step 4: Increase Result by 8

Add 8 to the result from Step 3. So, \( 30 + 8 = 38 \).

Key Concepts

Understanding Quotients

A quotient is simply the result you obtain when you divide one number by another. In mathematics, the division operation helps to split things into equal parts or groups. Think of it as sharing equally or distributing evenly among participants. In our exercise, we were tasked to find the quotient of 50 and \( \frac{5}{3} \). This part of the task involved dividing these two numbers. However, because the second number is a fraction, the process incorporates understanding more about how fractions work in division.

The Reciprocal in Division

When it comes to dividing by a fraction, the simplest method is to multiply by its reciprocal. But what exactly is a reciprocal? The reciprocal of a fraction is obtained by swapping its numerator and its denominator. For instance, the reciprocal of \( \frac{5}{3} \) is \( \frac{3}{5} \). To divide 50 by \( \frac{5}{3} \) means essentially multiplying 50 by \( \frac{3}{5} \). This step converts the complexity of a division operation into a more straightforward multiplication task, making it easier to compute.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This means you perform all operations such as addition, subtraction, multiplication, or division until you can't simplify them any further. In our example, after dividing 50 by \( \frac{5}{3} \) and finding the result to be 30, the task wasn't concluded yet. We needed to further simplify by performing an additional step—adding 8. This resulted in 38, which is the most simplified form of our initial expression. As a result, simplifying expressions combines both numerical operations and logical sequence of actions to reach the final simplified answer.

• Break down the expression into easily manageable operations.
• Compute each operation methodically and carefully.
• Ensure you follow any additional instructions like "increasing by a number."

Ensuring each part is correct leads to an accurate and simplified answer. The main aim is always clarity and simplicity.