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Chapter 5: Problem 110

Perform the indicated operations. $$0 . \overline{5} \div 0 . \overline{2}-0.75$$

Short Answer

Expert verified
1.75

Step by step solution

01

Convert repeating decimals to fractions

First, convert the repeating decimals into fractions. \ \( 0. \overline{5} = \frac{5}{9} \) and \( 0. \overline{2} = \frac{2}{9} \)
02

Divide fractions

Divide the fractions \( \frac{5}{9} \div \frac{2}{9} \). To divide fractions, multiply by the reciprocal of the divisor: \( \frac{5}{9} \times \frac{9}{2} = \frac{5 \times 9}{9 \times 2} = \frac{45}{18} = \frac{5}{2} \).
03

Simplify the result

Simplify the fraction \( \frac{5}{2} \) to a decimal, which is \( 2.5 \).
04

Subtract 0.75

Subtract 0.75 from 2.5: \ \( 2.5 - 0.75 = 1.75 \)

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Key Concepts

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

repeating decimals to fractions
Repeating decimals can often seem tricky, but there's a systematic way to convert them into fractions. Let's take a repeating decimal like 0.\bar{5}.Here鈥檚 how you do it:
  • First, you need to set the repeating decimal as a variable: Let鈥檚 call it \(x\). So, \(x = 0.\bar{5}\).
  • Next, multiply both sides of the equation by a power of 10 such that the decimal part repeats after the decimal point. For \(0.\bar{5}\), you only need to multiply by 10: \(10x = 5.\bar{5}\).
  • Subtract the original equation from this new one: \(10x - x = 5.\bar{5} - 0.\bar{5}\). This simplifies to \(9x = 5\).
  • Finally, solve for \(x\) by dividing both sides by 9: \(x = \frac{5}{9}\).
Using this method for \(0.\bar{2}\), you get \(x = \frac{2}{9}\). So, now you have:\(0.\bar{5} = \frac{5}{9}\) and \(0.\bar{2} = \frac{2}{9}\).
dividing fractions
Dividing fractions may seem complex, but with the reciprocal method, it becomes easy. Let's consider you want to divide:\(\frac{5}{9} \text{ by } \frac{2}{9}\).Here鈥檚 how to do it:
  • First, find the reciprocal of the divisor (the second fraction). The reciprocal of \(\frac{2}{9}\) is \(\frac{9}{2}\).
  • Next, multiply the first fraction by this reciprocal: \(\frac{5}{9} \times \frac{9}{2}\).
  • Multiply the numerators together and the denominators together: \(\frac{5 \times 9}{9 \times 2}\). This equals \(\frac{45}{18}\).
To further simplify, since 45 and 18 share a common factor of 9, divide both numerator and denominator by 9: \( \frac{45}{18} = \frac{5}{2}\). So \(\frac{5}{9} \text{ divided by } \frac{2}{9} = \frac{5}{2}\).
simplifying fractions
Simplifying fractions means reducing them to their lowest terms. In our previous example, we ended up with \(\frac{45}{18}\). Simplification works like this:
  • Find the greatest common factor (GCF) of the numerator and the denominator. For 45 and 18, the GCF is 9.
  • Divide both the numerator and the denominator by the GCF. Therefore, \( \frac{45}{18} \rightarrow \frac{45 \text{ 梅 9}}{18 \text{ 梅 9}} = \frac{5}{2}\).
A fraction is fully simplified when the numerator and denominator have no common factors other than 1. In this case, \(\frac{5}{2}\) is already in its simplest form.Remember, simplifying makes calculations easier and usually gives a result that's more understandable.
decimal subtraction
Subtracting decimals is just like subtracting whole numbers, but it's important to line up the decimal points. Let鈥檚 subtract 0.75 from 2.5.Here鈥檚 a step-by-step guide:
  • Write the numbers one under the other, ensuring the decimal points are aligned: \(2.500 \text{ and } 0.750\).
  • If the length of the decimals differs, you can append zeros to the shorter decimal to make it easier: becomes \(2.500 - 0.750\).
  • Subtract digit by digit starting from the rightmost side: \(0-0=0\),\(0-5=5\) (borrowing is necessary here),\(5-7=8\),\(and 2-0=2\).So, \(2.500 - 0.750 = 1.750\).
Thus, \(2.5 - 0.75 = 1.75\). This method works for any set of decimals you come across.

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Most popular questions from this chapter

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