Operations with Signed Rational Numbers: Practice

All three are negative, so the one farthest from zero is the least. Write $\tfrac{3}{4}$ as $0.75$; the absolute values are then $0.8$, $0.75$ and $0.7$.

$-0.8 < -0.75 < -0.7$

Subtracting is adding the opposite: $-\tfrac{2}{3} + \left(-\tfrac{1}{4}\right)$. Both are negative, so add the sizes over twelfths and keep the minus sign.

$\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$

So the answer is $-\tfrac{11}{12}$. The choice $-\tfrac{5}{12}$ subtracts the sizes, which is the rule for different signs.

Subtracting $-5.75$ is adding $5.75$, so this is $-3.2 + 5.75$. The signs differ, so subtract the smaller size from the larger.

$5.75 - 3.20 = 2.55$

The number $5.75$ is farther from zero, so the answer is positive, $2.55$.

Change the decimal to a fraction, $0.8 = \tfrac{4}{5}$, and compare the sizes over thirtieths: $\tfrac{5}{6} = \tfrac{25}{30}$ and $\tfrac{4}{5} = \tfrac{24}{30}$.

$\frac{25}{30} > \frac{24}{30}$

So $-\tfrac{5}{6}$ is farther from zero, which on the negative side means farther left: $-\tfrac{5}{6} < -0.8$.

The signs differ, so the product is negative. Multiply as whole numbers, then count decimal places: one in $1.2$ and two in $0.25$ make three.

$12 \times 25 = 300, \quad \text{so} \quad 1.2 \times 0.25 = 0.300 = 0.3$

So $(-1.2) \times 0.25 = -0.3$.

The signs differ, so the quotient is negative. For the size, multiply by the reciprocal of $\tfrac{5}{12}$.

$\frac{5}{6} \times \frac{12}{5} = \frac{60}{30} = 2$

So the quotient is $-2$. Check: $(-2) \times \tfrac{5}{12} = -\tfrac{10}{12} = -\tfrac{5}{6}$.

First write the mixed number as a fraction: $-1\tfrac{1}{2} = -\tfrac{3}{2}$. Its reciprocal is that fraction turned upside down, and it keeps the minus sign.

$\left(-\frac{3}{2}\right) \times \left(-\frac{2}{3}\right) = 1$

The product is $1$, so the reciprocal is $-\tfrac{2}{3}$. The choice $-2$ is the reciprocal of $-\tfrac{1}{2}$ alone.

The minus sign belongs to the whole mixed number, so $-1\tfrac{3}{4}$ is the opposite of $1 + \tfrac{3}{4}$.

$-1\tfrac{3}{4} = -\left(\frac{4}{4} + \frac{3}{4}\right) = -\frac{7}{4}$

The choice $-\tfrac{1}{4}$ comes from reading it as $-1 + \tfrac{3}{4}$, which is a different number.

No factor is zero, so count the negative factors. There are three, an odd count, so the product is negative.

$0.4 \times 0.5 = 0.2, \quad 0.2 \times 10 = 2$

So the product is $-2$.

A fall subtracts, so the new reading is $-4.5 - 2.75$, which is $-4.5 + (-2.75)$. Both are negative, so add the sizes and keep the sign.

$4.50 + 2.75 = 7.25$

The thermometer reads $-7.25$ degrees, colder than before, as a fall should leave it.

Division asks for the missing factor: what number times $-\tfrac{3}{7}$ gives $0$? Exactly one number does, and it is $0$.

$0 \times \left(-\frac{3}{7}\right) = 0$

So the quotient is $0$. Only dividing by zero is undefined; dividing zero by a nonzero number is fine.

The opposite of $-\tfrac{3}{8}$ is $\tfrac{3}{8}$, so the problem is $\tfrac{3}{8} - \tfrac{1}{2}$. Over eighths that is $\tfrac{3}{8} + \left(-\tfrac{4}{8}\right)$.

$\frac{4}{8} - \frac{3}{8} = \frac{1}{8}$

The number $-\tfrac{4}{8}$ is farther from zero, so the answer is $-\tfrac{1}{8}$.