Problem 112 - Fundamentals of Mathematics

Question

A full box contains 80 calculators. How many calculators are in \(\frac{1}{4}\) of a box?

Step-by-Step Solution

Answer

There are 20 calculators in \\(\frac{1}{4}\\) of a box.

Step 1: Understand the Problem

We start with a full box that contains 80 calculators. We need to find out how many calculators are in one fourth of this box, which means \(\frac{1}{4}\) of 80.

Step 2: Set Up the Expression

To find \(\frac{1}{4}\) of 80, we set up the expression as \(\frac{1}{4} \times 80\). This will help us calculate the number of calculators in a quarter of the box.

Step 3: Calculate the Value

Perform the multiplication: \(\frac{1}{4} \times 80 = 20\). This tells us that there are 20 calculators in a fourth of the box.

Step 4: Conclusion

We found that \(\frac{1}{4}\) of a box contains 20 calculators. Therefore, the number of calculators in a quarter of the box is 20.

Key Concepts

Multiplication

Multiplication is a fundamental concept in mathematics that helps us find the total number or quantity of something. In our exercise, we used multiplication to determine how many calculators are found in one-fourth of a box that originally holds 80 calculators. When multiplying fractions by whole numbers, like \(\frac{1}{4} \times 80\), it helps to initially think of multiplication as repeated addition. However, with fractions, you also scale down the whole, in this case, taking just a part or a fraction of it.

• To multiply a fraction by a whole number, you multiply the numerator (top number) of the fraction by the whole number, then divide by the denominator (bottom number).
• This approach helps to scale the original whole number down to the size according to the fraction.

In our specific scenario, multiplying involves scaling down from an entire box to just a fourth of it. Therefore, by finding \(\frac{1}{4} \times 80\), we effectively calculated one part of four equal sections of calculators that originally make up the full box.

Word Problems

Word problems can often appear daunting at first, but they simply describe real-world situations that require mathematical solutions. The key to solving word problems is understanding what is being asked and identifying the mathematical operation needed. In this exercise, the word problem describes how many calculators are in a portion of a box.To tackle word problems:

• Read the problem several times to understand the scenario.
• Identify key details, such as the total number from which parts are being calculated – here it was 80 calculators.
• Determine what portion or fraction of the total you need to find – hence, fraction multiplication.
• Translate these details into a mathematical expression or equation that allows for solving the problem, like using the expression \(\frac{1}{4} \times 80\).

By systematically breaking down the problem, you can better see how and why specific mathematical operations, such as multiplication or division, are needed and how they are applied to solve the problem.

Division

Division often goes hand in hand with multiplication, especially when dealing with fractions and portions. It is the process of distributing a number into parts. In our calculator box problem, \(\frac{1}{4} \) indicates division, as it asks us to divide the box into four equal parts.Key points about division in this context:

• The fraction \(\frac{1}{4}\) can be thought of as asking "what do you get when 80 is divided into four equal parts?"
• To understand division by a fraction, recognize it as multiplying by its reciprocal. Thus, dividing by 4 is also achieved by multiplying by \(\frac{1}{4}\).
• Division helps in determining exact portions of a whole, which is crucial in understanding diversely portioned quantities in real-life scenarios.

Combining division with multiplication enables us to solve problems involving fractions with more clarity, offering a structured method of analyzing quantities such as calculators in parts of boxes.