# Simple Problem

#### AnonProxy

##### New member
Hi,

Apologies this is a simple problem but yet I have forgotten how to solve since school days .

The problem is :

(x/6) = (x/2) - (4/3)

The explanation to the answer is:

 (x/6) = (x/2) - (4/3) / Find a common denominator. (2x/12) = (6x/12) - (16/12) / Multiply both sides by 12 2x = 6x - 16 / Subtract 6x from each side. -4x = -16 / Divide each side by -4. x = -16/-4 x = 4
Sorry I don't understand the explanation. Where is the 16 from?

Thanks

#### JeffM

##### Elite Member
Hi,

Apologies this is a simple problem but yet I have forgotten how to solve since school days .

The problem is :

(x/6) = (x/2) - (4/3)

The explanation to the answer is:

 (x/6) = (x/2) - (4/3) / Find a common denominator. (2x/12) = (6x/12) - (16/12) / Multiply both sides by 12 2x = 6x - 16 / Subtract 6x from each side. -4x = -16 / Divide each side by -4. x = -16/-4 x = 4
Sorry I don't understand the explanation. Where is the 16 from?

Thanks
$$\displaystyle \dfrac{x}{6} = \dfrac{x}{2} - \dfrac{4}{3}.$$

I have no idea why they suggest multiplying by 12. It's easier and just as effective to multiply by 6. In any case, here is a step by step following their suggestion.

$$\displaystyle \dfrac{x}{6} = \dfrac{x}{2} - \dfrac{4}{3} \implies 12 * \dfrac{x}{6} = 12 * \left(\dfrac{x}{2} - \dfrac{4}{3}\right).$$ Multiplying both sides of the equation by 12 as suggested.

$$\displaystyle 12 * \dfrac{x}{6} = 12 * \left(\dfrac{x}{2} - \dfrac{4}{3}\right) \implies \dfrac{12x}{6} = \dfrac{12x}{2} - \dfrac{12 * 4}{3}$$

$$\displaystyle \implies 2x = 6x - \dfrac{48}{3}$$

$$\displaystyle \implies 2x = 6x - 16$$

$$\displaystyle \implies 2x + 16 = 6x - 16 + 16$$

$$\displaystyle \implies 2x + 16 = 6x$$

$$\displaystyle \implies 6x = 2x + 16$$

$$\displaystyle \implies - 2x + 6x = - 2x + 2x + 16$$

$$\displaystyle \implies 6x - 2x = 16$$

$$\displaystyle \implies 4x = 16$$

$$\displaystyle \implies \dfrac{4x}{4} = \dfrac{16}{4}$$

$$\displaystyle \implies x = 4.$$

#### HallsofIvy

##### Elite Member
You have the equation $$\displaystyle \frac{x}{6}= \frac{x}{2}- \frac{4}{3}$$
A common denominator, a number evenly divisible by 6, 2, and 3, is 12.
(Actually, the least common denominator is 6.)

Multiplying both sides of the equation by 12 gives
$$\displaystyle \frac{x}{6}(12)= \frac{x}{2}(12)- \frac{4}{3}(12)$$
12/6= 2 so $$\displaystyle \frac{x}{6}(12)= 2x$$
12/2= 6 so $$\displaystyle \frac{x}{2}(12)= 6x$$
12/3= 4 so $$\displaystyle \frac{4}{3}(12)= 4(4)= 16$$

2x= 6x- 16. Subtracting 6x from both sides gives -4x= -16 so x= -16/-4= 4.

As I pointed out above, the least common denominator is 6, also evenly divisible by 6, 2, and 3.
If we multiplied both sides of the equation by 6 instead of 12, we get
$$\displaystyle \frac{x}{6}(6)= \frac{x}{2}(6)- \frac{4}{3}(6)$$
6/6= 1 so $$\displaystyle \frac{x}{6}(6)= x$$
6/2= 3 so $$\displaystyle \frac{x}{2}(6)= 3x$$
6/3= 2 so $$\displaystyle \frac{4}{3}(6)= 2(4)= 8$$

x= 3x- 8. Subtracting 3x from both sides gives -2x= -8 so x= -8/-2= 4 again.

That's how I would have done the problem.

Hah-Hah! I beat Denis by four minutes.

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Thanks