Question

How do you simplify \[\dfrac{{17 \cdot 4 + 17 \cdot 9 + 17 \cdot 7}}{{2 + 34 + 5 \cdot 34 + 3 \cdot 34}}\] ?

Answer 1

Hint: In order to solve this question, first of all we will take \[17\] as common in the numerator and \[34\] as common in the denominator from the last three terms. Then we will get the expression as \[\dfrac{{17\left( {4 + 9 + 7} \right)}}{{2 + 34\left( {1 + 5 + 3} \right)}}\] . After that we will add the terms in the numerator and denominator. Then we will perform multiplication and finally simplify further to get the required result.

Complete step-by-step answer:
We have given the expression as
\[\dfrac{{17 \cdot 4 + 17 \cdot 9 + 17 \cdot 7}}{{2 + 34 + 5 \cdot 34 + 3 \cdot 34}}\]
And we are asked to simplify it.
So, first of all take \[17\] as common from the numerator
\[ = \dfrac{{17\left( {4 + 9 + 7} \right)}}{{2 + 34 + 5 \cdot 34 + 3 \cdot 34}}\]
Now take \[34\] as common from the last three terms of the denominator
\[ = \dfrac{{17\left( {4 + 9 + 7} \right)}}{{2 + 34\left( {1 + 5 + 3} \right)}}\]
Now add the terms in the numerator as well as in the denominator
\[ = \dfrac{{17\left( {20} \right)}}{{2 + 34\left( 9 \right)}}\]
Now multiply the terms in the numerator as well as in the denominator
\[ = \dfrac{{340}}{{2 + 306}}\]
Now add the terms in the denominator
\[ = \dfrac{{340}}{{308}}\]
Now perform prime factorization in the numerator and denominator both
\[ = \dfrac{{2 \times 2 \times 5 \times 17}}{{2 \times 2 \times 7 \times 11}}\]
Now eliminate like terms from both numerator and denominator
\[ = \dfrac{{5 \times 17}}{{7 \times 11}}\]
Now multiply the terms
\[ = \dfrac{{85}}{{77}}\]
which is the required answer.
Hence, the value of the given expression \[\dfrac{{17 \cdot 4 + 17 \cdot 9 + 17 \cdot 7}}{{2 + 34 + 5 \cdot 34 + 3 \cdot 34}}\] is equal to \[\dfrac{{85}}{{77}}\]

Note: This question can also be solved by using the PEMDAS rule.
PEMDAS is a set of rules for performing operations in an arithmetic expression. The full form of PEMDAS is as follow-
P- Parentheses
E- Exponents
MD- Multiplication and Division (perform left to right based on whichever operation comes first)
AS- Addition and subtraction (perform left to right based on whichever operation comes first)

Now in our problem, the given expression is \[\dfrac{{17 \cdot 4 + 17 \cdot 9 + 17 \cdot 7}}{{2 + 34 + 5 \cdot 34 + 3 \cdot 34}}\]
According to PEMDAS rule, first we perform multiplication, therefore, we get
\[ = \dfrac{{68 + 153 + 119}}{{2 + 34 + 170 + 102}}\]
Now we will perform addition, so we have
\[ = \dfrac{{340}}{{308}}\]
Now cancelling the common factors, we get the final result as
\[ = \dfrac{{85}}{{77}}\]