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Chapter 4: Problem 234

In the following exercises, simplify. $$\frac{8 \cdot 3+2 \cdot 9}{14+3}$$

Short Answer

Expert verified
The simplified form is $$\frac{42}{17}$$

Step by step solution

01

Simplify the Numerator

Evaluate the expression in the numerator: $$8 \times 3 + 2 \times 9$$
02

Perform Multiplication

Multiply the numbers in the numerator: $$8 \times 3 = 24$$ $$2 \times 9 = 18$$
03

Add the Results

Add the results of the multiplications in the numerator: $$24 + 18 = 42$$ Now, the numerator is simplified to 42.
04

Simplify the Denominator

Evaluate the expression in the denominator: $$14 + 3$$
05

Perform Addition in the Denominator

Add the numbers in the denominator: $$14 + 3 = 17$$ Now, the denominator is simplified to 17.
06

Divide the Numerator by the Denominator

Divide the simplified numerator by the simplified denominator: $$\frac{42}{17}$$

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Key Concepts

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

Numerator Simplification
The numerator is the top part of a fraction. To simplify it, we first break down any operations like addition or multiplication. In our example, we start with the expression in the numerator:
  • $$8 \times 3 + 2 \times 9$$
We need to handle the multiplication parts first. So, we do the following steps:
  • $$8 \times 3 = 24$$
  • $$2 \times 9 = 18$$
After performing these multiplications, the next step is to add these results together:
  • $$24 + 18$$
This gives us:
  • $$42$$
Now, our numerator is simplified to 42. The key steps are: multiply first, then add the results. This way, we simplify our calculations effectively.
Denominator Simplification
The denominator is the bottom part of a fraction. Simplifying it is often straightforward but just as crucial. For the given exercise, we have the expression:
  • $$14 + 3$$
Unlike the numerator, there's no multiplication here, just a simple addition. We add these numbers together:
  • $$14 + 3 = 17$$
Hence, the denominator simplifies to 17. Simplifying the denominator means handling the arithmetic operations as per their natural order, which is pretty basic but ensures accuracy in the final fraction form.
Division of Fractions
The final step in simplifying a fraction involves division. After simplifying both the numerator and the denominator, we get a new fraction to work with:
  • $$\frac{42}{17}$$
This is now a simplified fraction. We can't simplify it further because 42 and 17 don't have any common factors. Division of fractions generally means just writing your simplified numerator over your simplified denominator and checking if further simplification is possible. If it isn鈥檛, like in our case, you have your final answer.
Remember that keeping things in order鈥攕implify numerator first, then the denominator, and finally, write down the fraction鈥攈elps you avoid mistakes.

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Most popular questions from this chapter

Convert the mixed number to an improper fraction. $$ 3 \frac{2}{7} $$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(x-\frac{2}{5}\) when (a) \(x=\frac{3}{5} \quad\) (b) \(x=-\frac{3}{5}\)

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