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Chapter 4: Problem 39

Subtract. $$ \begin{array}{r} 4 \frac{7}{10} \\ -2 \frac{1}{10} \\ \hline \end{array} $$

Short Answer

Expert verified
The result of the subtraction is \(2 \frac{3}{5}\).

Step by step solution

01

Convert to improper fractions

First, turn the mixed numbers into improper fractions. For the first number, \(4 \frac{7}{10}\), multiply the whole number \(4\) by the denominator \(10\) and add the numerator \(7\). This gives \(4 \times 10 + 7 = 47\). So, \(4 \frac{7}{10}\) becomes \(\frac{47}{10}\).For the second number, \(2 \frac{1}{10}\), multiply the whole number \(2\) by the denominator \(10\) and add the numerator \(1\). This gives \(2 \times 10 + 1 = 21\). So, \(2 \frac{1}{10}\) becomes \(\frac{21}{10}\).
02

Subtract the improper fractions

Now, subtract the improper fractions found in Step 1. Subtract \(\frac{21}{10}\) from \(\frac{47}{10}\):\[\frac{47}{10} - \frac{21}{10} = \frac{47 - 21}{10} = \frac{26}{10}\]
03

Simplify the result

The fraction \(\frac{26}{10}\) can be simplified. Find the greatest common divisor of \(26\) and \(10\), which is \(2\). Divide both the numerator and the denominator by \(2\):\[\frac{26}{10} = \frac{26 \div 2}{10 \div 2} = \frac{13}{5}\]
04

Convert back to a mixed number

Convert the improper fraction \(\frac{13}{5}\) back into a mixed number. Divide \(13\) by \(5\), which goes \(2\) times with a remainder of \(3\). So, \(\frac{13}{5}\) is the same as \(2 \frac{3}{5}\).

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Key Concepts

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

Improper Fractions
Improper fractions can be a bit tricky if you're new to them, but they are quite straightforward once you get the hang of it. An improper fraction is simply a fraction where the numerator (the top number) is larger than or equal to the denominator (the bottom number). This means the fraction represents a number greater than 1.
  • Example: In the fraction \( \frac{9}{4} \), the numerator 9 is larger than the denominator 4, hence it's an improper fraction.
  • The whole point of converting mixed numbers to improper fractions is to make operations like addition and subtraction straightforward, especially when the denominators are the same.
When converting a mixed number to an improper fraction, you multiply the whole number by the fraction's denominator and add it to the numerator. For example, if we have \(4 \frac{7}{10}\), we calculate:
  • Multiply the whole number 4 by the denominator 10, giving 40.
  • Add it to the numerator 7, leading to 47.
  • Thus, \(4 \frac{7}{10}\) converts to \(\frac{47}{10}\).
Mixed Numbers
Mixed numbers consist of a whole number and a fraction. They are very common in everyday mathematics because they are intuitive and easy to visualize. For instance, when you see \(2 \frac{3}{4}\), it's easy to imagine two whole pizzas and \(\frac{3}{4}\) of another pizza.
  • Mixed numbers provide a simple way to express the sum of whole numbers and fractions.
  • They are particularly useful when dealing with measurements, like feet and inches, or hours and minutes.
      • To convert an improper fraction back into a mixed number, you perform division to determine how many whole parts there are. Let's take the improper fraction \(\frac{13}{5}\). You divide 13 by 5:
      • The quotient is 2, which becomes the whole number of the mixed number.
      • The remainder is 3, which remains the numerator of the fraction.
      • Thus, \(\frac{13}{5}\) converts to \(2 \frac{3}{5}\).
Fraction Simplification
Simplifying fractions is a crucial skill in algebra and arithmetic, making it easier to manage and understand fractions. To simplify a fraction means to transform it into its simplest form. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD).
  • The GCD of two numbers is the largest number that divides both of them without leaving a remainder.
  • For instance, to simplify \(\frac{26}{10}\), we find the GCD of 26 and 10, which is 2.
  • Dividing both by 2 gives us the simplified fraction \(\frac{13}{5}\).
Simplifying fractions helps in comparing, adding, and subtracting them more easily. The simplified fraction \(\frac{13}{5}\) represents the same value as \(\frac{26}{10}\) but in a more convenient form.

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Most popular questions from this chapter

Write each mixed number as an improper fraction. See Example 20. $$10 \frac{14}{27}$$

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{37}{8}$$

Simplify. See Section 1.7. $$3^{4}$$

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{65}{12}$$

Perform the indicated operations. $$ 11 \frac{7}{8}-13 \frac{5}{6} $$
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