Equations with Fractions

A third of x

$\frac{x}{3}=4$x/3 = 4 says that a third of x is 4. The fraction is in the way: x has been divided by 3, and you want x on its own.

If one third of x is 4, then all three thirds together are 3 × 4 = 12. So x = 12.

x is cut into three equal thirds, and one third is 4. The three thirds make 3 × 4 = 12.

Multiply both sides by the denominator

Multiplication undoes division. x has been divided by 3, so multiply both sides by 3: $\frac{x}{3}\times 3=4\times 3$x/3 × 3 = 4 × 3, which gives x = 12. The balance stays level because both pans were multiplied by the same number.

Check: 12 ÷ 3 = 4.

Multiply both sides of $\frac{x}{3}=4$x/3 = 4 by 3, and the pans hold x and 12: x = 12.

Multiply every term

When the equation has more than one term on a side, multiply every term. In $\frac{x}{3}+2=6$x/3 + 2 = 6, multiply both sides by 3: $\frac{x}{3}\times 3=x$x/3 × 3 = x, 2 × 3 = 6 and 6 × 3 = 18. The equation becomes x + 6 = 18, so x = 12.

The 2 must be multiplied as well, even though it is not a fraction. The whole left side is multiplied by 3, and the 2 is part of it. Check: $\frac{12}{3}+2=4+2=6$12/3 + 2 = 4 + 2 = 6.

Two different denominators

In $\frac{x}{3}+\frac{x}{4}=7$x/3 + x/4 = 7 there are two fractions, one over 3 and one over 4. Multiply by a number that both 3 and 4 divide into. The lowest common multiple of 3 and 4 is 12, so multiply every term by 12.

$\frac{x}{3}\times 12=4x$x/3 × 12 = 4x, because 12 ÷ 3 = 4. $\frac{x}{4}\times 12=3x$x/4 × 12 = 3x, because 12 ÷ 4 = 3. And 7 × 12 = 84. The equation becomes 4x + 3x = 84, so 7x = 84 and x = 12.

Check: $\frac{12}{3}+\frac{12}{4}=4+3=7$12/3 + 12/4 = 4 + 3 = 7.

Cut x into 12 equal pieces. A third of x is 4 pieces and a quarter of x is 3 pieces, so $\frac{x}{3}+\frac{x}{4}$x/3 + x/4 is 7 twelfths of x.

A whole expression over the line

In $\frac{\left(x+2\right)}{5}=3$(x + 2)/5 = 3, the whole of x + 2 is divided by 5. Multiply both sides by 5: x + 2 = 15. Then subtract 2 from both sides: x = 13. Check: $\frac{\left(13+2\right)}{5}=\frac{15}{5}=3$(13 + 2)/5 = 15/5 = 3.

The usual mistakes

Leaving out a term that is not a fraction. In $\frac{x}{3}+2=6$x/3 + 2 = 6, multiplying the $\frac{x}{3}$x/3 and the 6 by 3 but not the 2 gives x + 2 = 18, so x = 16. The check fails: $\frac{16}{3}+2$16/3 + 2 is not 6. Every term on both sides is multiplied.

Dividing instead of multiplying. $\frac{x}{3}=4$x/3 = 4 does not mean $x=4/3$x = 4 / 3. x has already been divided by 3, and dividing again moves further away from x.