# 8. Equations Involving Fractions

In this section, we can find the solution easily by
multiplying throughout by the **lowest common
denominator.**

### Example 1

An aquarium can be filled by one hose in 7 minutes and a second thinner hose in 10 minutes.

How long will it take to fill the tank if both hoses operate together?

Answer

Since each hose makes the filling time **less**, we have
to add the reciprocals together and take the reciprocal of the
result.

We need to use:

`1/T=1/T_1+1/T_2`

So we have:

`1/T=1/7+1/10`

`1/T=(10+7)/70=17/70`

So

`T=70/17=4.1176`

So it will take 4.1 minutes to fill the tank with both hoses operating together.

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### Example 2

For 2 resistors with resistances
*R*_{1}and
*R*_{2} in parallel, the combined
resistance *R* is given by:

`1/R=1/R_1+1/R_2`

For a particular circuit, the combined resistance
*R* was found to be 4 ohms (Ω),
and *R*_{1}= 10
Ω. Find
*R*_{2}.

Answer

We have:

`1/4=1/10+1/R_2`

We need to multiply throughout by the **lowest common
denominator: 20***R*_{2}

`1/4=1/10+1/R_2`

`(20R_2)/4=(20R_2)/10+(20R_2)/(R_2)`

`5R_2=2R_2+20`

`3R_2=20`

`R_2=20/3=6 2/3Omega`

"`Omega`" is the symbol for "ohms", the unit of resistance.

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### Example 3

A car averaged 30 km/h going from home to work and 40 km/h on the return journey. If the total time for the two journeys is 50 minutes, how far is it from home to work?

Answer

Let the length of the journey from home to work be *x* km.

Recall that

`text(speed) = text(distance)/text(time)`

So

`text(time) = text(distance)/text(speed)`

We must use the same time units throughout. We will use hours.

Now

` 50\ text(minutes)=50/60=5/6text(hours)`

In the forward journey, the car's time was `x/30` hours.

For the return journey, the time was `x/40` hours.

The total time was `x/30+x/40=5/6\ text(hours)`

So `(4x+3x)/120=(7x)/120=5/6`

This gives us `7x=(5xx120)/6=100`

That is `x=100/7=14.286`

So the distance from home to work is 14.3 km.

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