Problem 49 - Prealgebra

Question

Simplify each expression as much as possible. If the quotient of 18 and \(\frac{3}{5}\) is increased by \(10,\) what number results?

Step-by-Step Solution

Answer

The resulting number is 40.

Step 1: Divide 18 by \( \frac{3}{5} \)

To divide by a fraction, multiply by its reciprocal. The expression \( \frac{18}{\frac{3}{5}} \) can be rewritten as \( 18 \times \frac{5}{3} \).

Step 2: Multiply 18 by the Reciprocal

Perform the multiplication: \( 18 \times \frac{5}{3} = \frac{18 \times 5}{3} = \frac{90}{3} \).

Step 3: Simplify the Fraction

Divide the numerator by the denominator: \( \frac{90}{3} = 30 \). The result of the division is 30.

Step 4: Add 10 to the Quotient

Now, add 10 to the result from Step 3. So, \( 30 + 10 = 40 \).

Key Concepts

Fraction Division

Fraction division can seem confusing at first, but it becomes much simpler once you understand the core idea. Dividing by a fraction is the same as multiplying by its reciprocal.

• First, identify the fraction you're dividing by.
• Find its reciprocal by flipping the numerator and the denominator.
• Then, instead of dividing, multiply by this reciprocal.

This is an essential concept in prealgebra that makes fraction division much easier to handle. So instead of directly dividing, switch to the operation of multiplication.

Multiplication by Reciprocals

The idea of multiplying by reciprocals is a key process in fraction division. Once you have found the reciprocal of the fraction you are dividing by, you essentially "flip" the fraction.

• For example, the reciprocal of \( \frac{3}{5} \) is \( \frac{5}{3} \).

By multiplying by the reciprocal, you turn the division problem into a multiplication one. It simplifies the calculation process significantly. Understanding this trick can save you time and effort when working with fractions.

Simplifying Fractions

Simplifying fractions is the process of making the fraction as simple as possible. This means reducing it to its smallest form.

• You do this by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• In our step-by-step solution, we went from \( \frac{90}{3} \) to 30 by dividing 90 and 3 by their GCD, which is 3.

Simplified fractions are often easier to work with, and they make mathematical expressions cleaner and easier to interpret.

Addition

Addition seems simple, but it’s important to do it correctly in multi-step problems. After solving a division or multiplication problem, you might need to add values to get the final answer.

• For example, after simplifying \( \frac{90}{3} \) to 30, we added 10 to get 40.

Ensure you carefully perform any subsequent operations like addition to achieve the correct result. Each operation builds on the previous one, leading you to the final solution.

Numerator and Denominator

In any fraction, the numerator and the denominator play crucial roles. The numerator is the number above the line, representing the parts we have, while the denominator is below the line, showing how many parts make a whole.

• When dividing and simplifying, always keep clear which number is the numerator and which is the denominator.
• The position of these numbers will determine the fraction's value and make it clear what operations to perform.

Knowing the role of each helps you handle fractions properly during calculations.

Basic Arithmetic Operations

Basic arithmetic operations such as addition, subtraction, multiplication, and division are the backbone of mathematics. These operations allow us to solve a wide range of problems and simplify expressions.

• Understanding how to transform a division into a multiplication using reciprocals is crucial.
• Simplifying and then adding further calculations shows how these operations interlace.

Mastery of these basic operations is necessary to smoothly progress through more complex math problems.