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Chapter 7: Problem 62

In the following exercises, simplify. $$ \left(\frac{5}{6}+\frac{8}{15}\right)+\frac{7}{15} $$

Short Answer

Expert verified
\frac{11}{6}

Step by step solution

01

Simplify the Expression Inside the Parentheses

First, add the fractions \(\frac{5}{6} \text{ and } \frac{8}{15}\). To do this, find a common denominator. The least common multiple of 6 and 15 is 30. Convert each fraction to have a denominator of 30: \(\frac{5}{6} = \frac{25}{30}\) and \(\frac{8}{15} = \frac{16}{30}\). Now add the fractions: \(\frac{25}{30} + \frac{16}{30} = \frac{41}{30}\).
02

Add the Result to the Outside Fraction

Now, add the resulting fraction from Step 1 to \(\frac{7}{15}\). First, we need to change \(\frac{7}{15}\) to have the same denominator as \(\frac{41}{30}\). Convert \(\frac{7}{15} = \frac{14}{30}\). Now add the fractions: \(\frac{41}{30} + \frac{14}{30} = \frac{55}{30}\).
03

Simplify the Final Fraction

Simplify \(\frac{55}{30}\) by finding the greatest common divisor (GCD) of the numerator and the denominator, which is 5. So, \(\frac{55}{30} = \frac{11}{6}\).

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Key Concepts

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

Common Denominator
To add fractions with different denominators, you need a common denominator. The common denominator is the least common multiple (LCM) of the two denominators.
For example, to add \(\frac{5}{6}\) and \(\frac{8}{15}\), find the LCM of 6 and 15. List the multiples:
  • Multiples of 6: 6, 12, 18, 24, 30...
  • Multiples of 15: 15, 30, 45...
The smallest common multiple is 30. Thus, 30 is our common denominator.
Next, convert each fraction. For \(\frac{5}{6}\), multiply the numerator and denominator by 5 to get \(\frac{25}{30}\). For \(\frac{8}{15}\), multiply the numerator and denominator by 2 to get \(\frac{16}{30}\).
Now that they have the same denominator, you can add: \(\frac{25}{30} + \frac{16}{30} = \frac{41}{30}\).
Simplifying Fractions
After you add or subtract fractions, the resulting fraction may not be in its simplest form. To simplify, divide the numerator and the denominator by their greatest common divisor (GCD).
For instance, after adding our fractions, we had \(\frac{55}{30}\). To simplify this, find the GCD of 55 and 30.
The factors of 55 are 1, 5, 11, and 55. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The largest common factor is 5.
Divide both the numerator and denominator by 5: \(\frac{55 \div 5}{30 \div 5} = \frac{11}{6}\). Thus, \(\frac{55}{30}\) simplifies to \(\frac{11}{6}\).
Greatest Common Divisor
The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder.
You can find the GCD by listing out the factors of each number and picking the highest one they share.
For example, find the GCD of 55 and 30.
List the factors of each number:
  • Factors of 55: 1, 5, 11, 55
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The highest common factor is 5. This means 5 is the GCD of 55 and 30.
Knowing the GCD helps you simplify fractions by dividing the numerator and the denominator by this common divisor.

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Most popular questions from this chapter

In the following exercises, evaluate each expression for the given value. If \(k=65,\) evaluate: $$ (a)\frac{4}{9}\left(\frac{9}{4} k\right) $$ $$ (b)\left(\frac{4}{9} \cdot \frac{9}{4}\right) k $$

In the following exercises, convert the Fahrenheit to degrees Celsius. Round to the nearest tenth. $$ 64^{\circ} \mathrm{F} $$

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In the following exercises, identify whether each example is using the identity of addition or multiplication. $$ -35(1)=-35 $$

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