Operations with Signed Rational Numbers: Practice

The opposite of a number sits the same distance from zero, on the other side. The number $-\tfrac{3}{4}$ is three quarters of a unit left of zero, so its opposite is three quarters of a unit right of zero.

$-\left(-\tfrac{3}{4}\right) = \tfrac{3}{4}$

Turning the fraction upside down gives a reciprocal, which is a different idea.

The decimal $0.7$ has one decimal place, so it counts tenths: seven tenths. Its opposite, $-0.7$, is negative seven tenths.

$-0.7 = -\frac{7}{10} = \frac{-7}{10}$

The choice $\frac{-7}{100}$ is $-0.07$, and $\frac{7}{10}$ has lost the sign.

The number $2.5$ lies halfway between $2$ and $3$. Its opposite is the mirror image across zero, so it lies halfway between $-3$ and $-2$.

$-3 < -2.5 < -2$

Absolute value is distance from zero. The number $-1.8$ sits $1.8$ units to the left of zero, and a distance is never negative.

$\lvert -1.8 \rvert = 1.8$

Both numbers are negative, so the one farther from zero is farther left and therefore smaller. The absolute values are $0.3$ and $0.5$, and $-0.5$ is the farther from zero.

$-0.5 < -0.3$

Read the other way round, $-0.3 > -0.5$.

Both numbers are negative, so add the sizes and keep the minus sign. Line up the points, padding $1.5$ with a zero.

$1.50 + 2.25 = 3.75$

So $-1.5 + (-2.25) = -3.75$.

The signs differ, so subtract the smaller size from the larger and take the sign of the number farther from zero.

$1.4 - 0.6 = 0.8$

The number $-1.4$ is farther from zero, so the answer is $-0.8$.

Subtracting is adding the opposite, so this is $\tfrac{1}{4} + \left(-\tfrac{1}{3}\right)$. Over twelfths the sizes are $\tfrac{3}{12}$ and $\tfrac{4}{12}$.

$\frac{4}{12} - \frac{3}{12} = \frac{1}{12}$

The number $-\tfrac{1}{3}$ is farther from zero, so the answer is $-\tfrac{1}{12}$.

One factor is negative and one is positive, so the product is negative. Multiply the sizes: half of $6$ is $3$.

$(-0.5) \times 6 = -3$

Both factors are negative, so the signs match and the product is positive. Multiply the tops and the bottoms for the size.

$\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}$

The signs match, so the quotient is positive. For the size, slide both points one place right so the divisor is a whole number.

$2.4 \div 0.6 = 24 \div 6 = 4$

Check: $4 \times (-0.6) = -2.4$.

A fraction is negative when its numerator and denominator have different signs, and its size is the size of the top over the size of the bottom.

$\frac{5}{-8} = -\frac{5}{8}$

In $\frac{-5}{-8}$ the signs match, so it is positive, and $\frac{8}{-5}$ is upside down.