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Chapter 1: Problem 15

\text { What is } \frac{3}{8}-\frac{1}{2}+\left(\frac{2}{4}\right)^{2} \text { ? }

Short Answer

Expert verified
The result is \( \frac{1}{8} \).

Step by step solution

01

Simplify the expression

First, rewrite the fractions with a common denominator. Notice that the denominator 8 is a common multiple of 2 and 4. Rewrite all fractions in terms of eighths: \( \frac{3}{8} \), \( \frac{1}{2} = \frac{4}{8} \), and \( \left(\frac{2}{4}\right)^{2} = \frac{1}{2}^{2} = \frac{1}{4} = \frac{2}{8} \).
02

Simplify the squared term

Before combining, simplify \( \left(\frac{2}{4}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} = \frac{2}{8} \).
03

Perform the addition and subtraction

Now, combine all the terms: \( \frac{3}{8} - \frac{4}{8} + \frac{2}{8} \).
04

Calculate the result

Perform the operations in order: \( \frac{3}{8} - \frac{4}{8} = -\frac{1}{8} \), and then add \( \frac{2}{8} \) to get \( 0 + \frac{1}{8} = \frac{1}{8} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When working with fractions, finding a common denominator is crucial for operations like addition and subtraction. Let's explore why it's essential and how to do it efficiently.

Fractions represent parts of a whole. To add or subtract them, they must have the same denominator. The denominator tells us into how many parts that whole is divided. For example, in the problem given, the fractions are \( \frac{3}{8} \), \( \frac{1}{2} \), and \( \left( \frac{2}{4} \right)^2 \).
  • The first step is to identify or create common denominators. Here, the number 8 was chosen because it's a common multiple of 2 and 4, the other denominators.
  • Convert each fraction to have this denominator: \( \frac{1}{2} \) becomes \( \frac{4}{8} \) and \( \left( \frac{2}{4} \right)^2 \) simplifies to \( \frac{2}{8} \).
Once each fraction has the same denominator, they can be added or subtracted directly. This makes calculations straightforward, as we only need to handle the numerators.
Simplification
Simplification is the process of making a mathematical expression more manageable by reducing it to its simplest form. This often involves reducing fractions or handling exponents.

In the exercise, we encounter the term \( \left( \frac{2}{4} \right)^2 \). Before performing arithmetic operations, simplifying this term is necessary:
  • First, \( \frac{2}{4} \) simplifies to \( \frac{1}{2} \) since both the numerator and denominator are divisible by 2.
  • Next, squaring \( \frac{1}{2} \) yields \( \frac{1}{4} \).
  • Then, convert \( \frac{1}{4} \) to eighths, resulting in \( \frac{2}{8} \).
Simplification eliminates complexity, providing simpler numbers for arithmetic operations. This makes it easier to find accurate results.
Arithmetic with Fractions
Arithmetic with fractions involves addition, subtraction, multiplication, or division. Once denominators are the same, you can perform operations on numerators.

In the given example, after ensuring all terms have a common denominator of 8, the expression \( \frac{3}{8} - \frac{4}{8} + \frac{2}{8} \) is straightforward:
  • Start with subtraction: \( \frac{3}{8} - \frac{4}{8} = -\frac{1}{8} \).
  • Then, add \( \frac{2}{8} \): \( -\frac{1}{8} + \frac{2}{8} = \frac{1}{8} \).
In arithmetic with fractions, the operations only affect the numerators when denominators match. This consistency allows for clear mathematical reasoning and accurate calculation.

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Most popular questions from this chapter

Evaluate each expression and check your results with a calculator. a. \(\frac{1}{5}+\frac{3}{4}\) b. \(3^{2}+2^{4}\) c. \(\frac{2}{3}+\left(\frac{6}{5}\right)^{2}\) d. \(4^{3}-\frac{2}{5}\)

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The base of a triangle was recorded as \(18.3 \pm 0.1 \mathrm{~cm}\) and the height was recorded as \(7.4 \pm 0.1 \mathrm{~cm}\). These measurements indicate the measured value and an accuracy component. a. Use the formula \(A=0.5 b h\) and the measured values for base and height to calculate the area of the triangle. b. Use the smallest possible lengths for base and height to calculate an area. (a) c. Use the largest possible lengths for base and height to calculate an area. d. Use your answers to \(12 \mathrm{a}-\mathrm{c}\) to express the range of possible area values as a number \(\pm\) an accuracy component. \((A\)

Find the five-number summary for each data set. a. \(\\{5,5,8,10,14,16,22,23,32,32,37,37,44,45,50\\}\) b. \(\\{10,15,20,22,25,30,30,33,34,36,37,41,47,50\\}\) c. \(\\{44,16,42,20,25,26,14,37,26,33,40,26,47\\}\) (a) d. \(\\{47,43,35,34,32,21,17,16,11,9,5,5\\}\)
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