# 8. Equations Involving Fractions

In this section, we can find the solution easily by multiplying throughout by the lowest common denominator (LCD). This will simplify our equation and make it easier to solve.

We need to remember to multiply all terms in the equation (both sides of the equal sign) by the LCD, otherwise the final answer won't be correct.

## a. Algebraic types

### Example 1

Solve for x:

x/5+3/10 = 1/2

We first look at the denominators of the fractions and determine the lowest common denominator. In this case, it will be 10.

We multiply throughout by 10 and the result is:

10xx(x/5+3/10) = 10xx(1/2)

2x+3=5

Then, we just solve the simpler equation we've found.

Subtract 3 from both sides:

2x=2

Divide boths sides by 2, giving us the final answer:

x=1

### Example 2

Solve for x:

(2x)/3+2/5 = 8+x/2

Once again, we look at the denominators of the fractions and determine the lowest common denominator. In this case, it will be 30.

We multiply throughout by 30 and the result is:

30xx((2x)/3+2/5) = 30xx(8+x/2)

20x+12=240+15x

Subtracting 12 from both sides:

20x=228+15x

Substract 15x from both sides:

5x=228

Divide both sides by 5, giving us the final answer:

x=228/5=45.6

### Example 3

This next one has the variable in the denominator. We'll still need to find the LCD as before.

Solve for x:

4/x+1/3 = 7/(5x)

In this case, the lowest common denominator is 15x.

We multiply throughout by 15x and the result is:

15x xx(4/x+1/3) = 15x xx(7/(5x))

60+5x=21

Subtracting 60 from both sides:

5x=-39

Divide both sides by 5, giving us the final answer:

x=-39/5=-7.8

Continues below

## b. Word problems ### Example 4

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?

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.

### Related pages

You may find these useful:

For 2 resistors with resistances R1and R2 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 R1= 10 Ω. Find R2.

We have:

1/4=1/10+1/R_2

We need to multiply throughout by the lowest common denominator: 20R2

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.

### Example 6

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?

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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