How to find the lowest / least common denominator - GRE Quantitative Reasoning

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Question

What is the least common denominator of and ?

Answer

To find the least common denominator of two numbers, it is easiest first to factor them into prime factors:

$150=25\cdot6=5^2\cdot2\cdot3=2\cdot3\cdot5^2$

$175=5^2\cdot7$

Now, you need to compare each number and choose the case in which the prime factor has the highest power. Therefore, since and are found only in , you will select those. You can take "either" . Finally, the in is the largest factor of . Your LCD is found by multiplying all of these together:

$2\cdot3\cdot5^2\cdot7 = 1050$

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Question

Simplify:

$\frac{4}{35}+\frac{7}{30}$

Answer

To begin to solve this, you need to find the least common denominator of and . The easiest way to do this is to begin by factoring them into prime factors:

$35=5\cdot7$

$30=2\cdot3\cdot5$

The LCD is found by selecting the largest power for each factor across the two values. Therefore, you will take ,, and from and the from the . Your LCD is therefore:

$2\cdot3\cdot5\cdot7=210$ .

Now, apply this to your fractions:

$\frac{4}{35}\cdot\frac{6}{6}+\frac{7}{30}\cdot\frac{7}{7}=\frac{24}{210}+\frac{49}{210}=\frac{73}{210}$

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Question

In simplest form, $\frac{11}{15}+\frac{17}{20}=$

Answer

First, find the smallest number that both and will factor into, which is .

This means that the fist fraction should be multiplied by and the second should be multiplied by .

Therefore,

$\frac{11}{15}\cdot \frac{4}{4}+\frac{17}{20}\cdot\frac{3}{3}=\frac{44}{60}+\frac{51}{60}$ .

Then, add the numerators while the denomenators stay the same:

$\frac{44}{60}+\frac{51}{60}=\frac{44+51}{60}=\frac{95}{60}$ .

Then, reduce the fraction to its lowest terms:

$\frac{95}{60}=\frac{5\cdot 19}{5 \cdot 12}=\frac{19}{12}$ .

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