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Chapter 8: Problem 53

Calculate. $$ \frac{7}{15}+\frac{4}{25} $$

Short Answer

Expert verified
\( \frac{47}{75} \)

Step by step solution

01

Find the common denominator

The denominators are 15 and 25. The least common multiple (LCM) of 15 and 25 is 75.
02

Convert fractions to have the common denominator

Convert \(\frac{7}{15}\) to a fraction with denominator 75: \(\frac{7}{15} \times \frac{5}{5} = \frac{35}{75}\). Convert \(\frac{4}{25}\) to a fraction with denominator 75: \(\frac{4}{25} \times \frac{3}{3} = \frac{12}{75}\).
03

Add the fractions

Now add the fractions with the same denominator: \[ \frac{35}{75} + \frac{12}{75} = \frac{47}{75} \]
04

Simplify if necessary

Check if \(\frac{47}{75}\) can be simplified. The numerator 47 and the denominator 75 do not have any common factors other than 1, so \(\frac{47}{75}\) is already in its simplest form.

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Key Concepts

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

least common multiple
The least common multiple (LCM) is essential when adding fractions with different denominators. The LCM of two numbers is the smallest number that is a multiple of both. To find the LCM of 15 and 25:
  • List the multiples of 15: 15, 30, 45, 60, 75, ...
  • List the multiples of 25: 25, 50, 75, ...
The smallest number that appears in both lists is 75. Thus, the LCM of 15 and 25 is 75. Having the same denominator allows for straightforward fraction addition.
fraction addition
Adding fractions involves aligning them to a common denominator. In the exercise, the fractions are \(\frac{7}{15}\) and \(\frac{4}{25}.\) Using the LCM of 75 as a common denominator, we convert:
  • Multiply \(\frac{7}{15}\) by \(\frac{5}{5}\) to get \(\frac{35}{75}.\)
  • Multiply \(\frac{4}{25}\) by \(\frac{3}{3}\) to get \(\frac{12}{75}.\)
Now, both fractions have the same denominator: \(\frac{35}{75}\) and \(\frac{12}{75}.\) Adding these fractions is straightforward:
\[ \frac{35}{75} + \frac{12}{75} = \frac{47}{75} \] This unified fraction makes it easy to see the new numerator.
simplification of fractions
Simplifying a fraction involves reducing it to its smallest form, where the numerator and denominator have no common factors other than 1. For \(\frac{47}{75},\) we check:
  • 47 is a prime number
  • 75 factors into 3 * 5 * 5
Because 47 and 75 have no common factors, \(\frac{47}{75}\) is in its simplest form. In cases where simplification is possible, divide both the numerator and denominator by their greatest common divisor (GCD). This step ensures the fraction is easy to understand and use in further calculations.

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Most popular questions from this chapter

Complete. 1 min =________sec

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The biblical measure called a span was considered to be half of a cubit \((1\) cubit \(\approx 18\) in. see Exercise 47 ). The giant Goliath's height "was six cubits and a span" (Holy Bible, King James Version, 1 Sam. 17: 4 ). What was the height of Goliath in inches? in feet?
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