need help with algebra word problem

one number is 3 times as large as another. The sum of their reciprocals is 20/3. find the two numbers.

Question

need help with algebra word problem

one number is 3 times as large as another. The sum of their reciprocals is 20/3. find the two numbers.

mathstudent55 · Answer

Can you pick a variable for the smaller number?

anonymous · Answer

i honestly have no clue on how to do this >_<

mathstudent55 · Answer

Let's call the smaller number x.
The other number is 3 times larger, so it's 3x.
Ok so far?

anonymous · Answer

yes

mathstudent55 · Answer

Do you know what a reciprocal is?

anonymous · Answer

is it when you flip the numbers?

mathstudent55 · Answer

Correct.
Now we have two numbers, x and 3x.
When you add their reciprocals you get 20/3.

The reciprocal of x is $\dfrac{1}{x}$. 

The recipoprocal of 3x is $ \dfrac{1}{3x} $.

Now we add the reciprocals and set that equal to $ \dfrac{20}{3} $.

mathstudent55 · Answer

That means we have

$ \dfrac{1}{x} + \dfrac{1}{3x} = \dfrac{20}{3} $

Now we need to solve for x.

anonymous · Answer

how would i go about doing that? I'm sorry, math is really not my best subject

mathstudent55 · Answer

Since you have an equation with fractions, the way to get rid of all denominators is to multiply both sides by the LCD.

What is the LCD of x, 3x, and 3?

anonymous · Answer

would it be 3x? or 3?

mathstudent55 · Answer

3x is correct

mathstudent55 · Answer

Now we multiply both sides by 3x

$3x \left( \dfrac{1}{x} + \dfrac{1}{3x} \right) = 3x \cdot \dfrac{20}{3} $

$ \ 3x \cdot \dfrac{1}{x} + 3x \cdot \dfrac{1}{3x}  = 3x \cdot \dfrac{20}{3} $

$ \ 3\cancel x \cdot \dfrac{1}{\cancel x} + \cancel{3x} \cdot \dfrac{1}{\cancel{3x}}  = \cancel 3x \cdot \dfrac{20}{\cancel{3}} $

$3 + 1 = 20x $

Can you solve for x now?

anonymous · Answer

x=5?

mathstudent55 · Answer

Be careful.

$4 = 20x$

Divide both sides by 20

$ \dfrac{4}{20} = x$

$ x = \dfrac{1}{5} $

The smaller number is 1/5.
The larger number is 3 times larger, so it's 3/5.

The numbers are 1/5 and 3/5.

mathstudent55 · Answer

Now let's check.

3/5 is 3 times large than 1/5, so the larger number is indeed 3 times larger than the smaller number.

Now let's add the reciprocals of 1/5 and 3/5 to see if we get 20/3.

5 + 5/3 
= 15/3 + 5/3 
= 20/3
The sum checks out, so the answer is correct.

anonymous · Answer

i always have the hardest time with word problems, but thank you so much for your help!

mathstudent55 · Answer

You're welcome.
To translate a word problem into an equation, select a variable and follow the statements.