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Chapter 5: Problem 93

Perform the indicated operations. Round the answer to the nearest hundredth when necessary. $$(42.81-30.01) \div \frac{9}{2}$$

Short Answer

Expert verified
2.84

Step by step solution

01

Subtract inside the parentheses

First, perform the subtraction inside the parentheses: \[ 42.81 - 30.01 = 12.80 \]
02

Divide by the fraction

Next, divide the result by the fraction \( \frac{9}{2} \). Dividing by a fraction is the same as multiplying by its reciprocal: \[ 12.80 \times \frac{2}{9} \]
03

Perform the multiplication

Multiply \( 12.80 \) by \( \frac{2}{9} \) to get the final result: \[ 12.80 \times \frac{2}{9} = 2.8444 \]
04

Round the result

Round \( 2.8444 \) to the nearest hundredth: \[ 2.84 \]

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Key Concepts

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

Subtraction
Subtraction is one of the basic arithmetic operations. It involves taking away one number from another. In the given exercise, we start by subtracting 30.01 from 42.81. To perform this operation:
  • Align the decimal points of both numbers.
  • Subtract the digits starting from the rightmost side.
  • Borrow from the next left digit if necessary.

After subtraction, 42.81 - 30.01, we get 12.80.
Division by a Fraction
Dividing by a fraction can sound tricky, but it's simpler if we understand the concept of reciprocals. When we divide by a fraction, we actually multiply by the reciprocal of that fraction. The reciprocal of a fraction is created by swapping the numerator (the top number) and the denominator (the bottom number).

In our exercise, we need to divide 12.80 by \(\frac{9}{2}\). The reciprocal of \(\frac{9}{2}\) is \(\frac{2}{9}\). This changes our problem from division to multiplication:

\[ 12.80 \times \frac{2}{9} \]

This step makes it more manageable to solve.
Multiplication
Multiplication is another fundamental arithmetic operation. To multiply a number by a fraction, you multiply the number by the numerator and then divide by the denominator.

In the exercise, we multiply 12.80 by the reciprocal of \(\frac{9}{2}\), which gives us the new multiplication problem: \[ 12.80 \times \frac{2}{9} \]

Here鈥檚 how to do it:
  • Multiply 12.80 by 2 to get 25.60.
  • Divide 25.60 by 9 to get 2.8444.

So, 12.80 times \(\frac{2}{9}\) is 2.8444.
Rounding Numbers
Rounding is a way to simplify a number to its approximate value. It helps in making calculations easier and results more readable. To round a number to the nearest hundredth, follow these steps:

  • Identify the digits in the hundredths and thousandths places.
  • If the digit in the thousandths place is 5 or more, increase the digit in the hundredths place by 1.
  • If the digit in the thousandths place is less than 5, keep the digit in the hundredths place unchanged.

For our answer, 2.8444, the digit in the thousandths place is 4, which is less than 5. Therefore, we round down:

\[ 2.8444 \rightarrow 2.84 \]

So, the rounded result is 2.84.

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