Operations with Signed Rational Numbers: Practice

The fraction $\tfrac{2}{3}$ repeats as a decimal, so write the decimal as a fraction instead: $-0.25 = -\tfrac{1}{4}$. Over twelfths the sizes are $\tfrac{3}{12}$ and $\tfrac{8}{12}$.

$\frac{8}{12} - \frac{3}{12} = \frac{5}{12}$

The signs differ and $\tfrac{2}{3}$ is farther from zero, so the answer is positive, $\tfrac{5}{12}$.

No factor is zero. Two of the factors are negative, an even count, so the product is positive.

$\frac{2}{3} \times \frac{9}{10} \times \frac{5}{6} = \frac{90}{180} = \frac{1}{2}$

Write both mixed numbers as fractions: $-1\tfrac{1}{2} = -\tfrac{3}{2}$ and $2\tfrac{1}{4} = \tfrac{9}{4}$. The signs differ, so the quotient is negative.

$\frac{3}{2} \times \frac{4}{9} = \frac{12}{18} = \frac{2}{3}$

So the quotient is $-\tfrac{2}{3}$. The choice $-\tfrac{27}{8}$ multiplies instead of dividing.

A debt is a negative balance, so the total is $-4.5$ dollars, split into $6$ equal parts. A negative divided by a positive is negative.

$4.5 \div 6 = 0.75$

Each share is $-0.75$ dollars. Check: $6 \times (-0.75) = -4.5$.

All four are negative, so the greatest is the one closest to zero, the one with the smallest absolute value. Compare $0.3 = \tfrac{3}{10}$ with $\tfrac{1}{3}$ over thirtieths.

$\frac{3}{10} = \frac{9}{30} < \frac{10}{30} = \frac{1}{3}$

The other two sizes are $0.35$ and $0.4$, both more than $\tfrac{1}{3}$. So $0.3$ is the smallest size, and $-0.3$ is the greatest.

Work out the inside first. Since $\tfrac{1}{2} = 0.5$, the inside is $0.5 - 0.75$; the signs then differ and $0.75$ is the larger size.

$0.5 - 0.75 = -0.25$

The minus sign in front means the opposite of that, and the opposite of $-0.25$ is $0.25$.

No factor is zero. All five factors are negative, an odd count, so the product is negative.

$\frac{1}{2} \times 3 \times 0.2 \times 4 \times \frac{5}{6} = 1.5 \times 0.2 \times 4 \times \frac{5}{6} = 1$

So the product is $-1$.

With no zero factor, the product is negative exactly when the count of negative factors is odd. Out of three numbers, the odd counts are $1$ and $3$.

$(-1) \times 2 \times 3 = -6, \qquad (-1) \times (-2) \times (-3) = -6$

Both cases happen, so neither "exactly one" nor "all three" must be true.

The rise is the later reading minus the earlier one. Subtracting $-3.6$ is adding $3.6$.

$2.1 - (-3.6) = 2.1 + 3.6 = 5.7$

It rose $5.7$ degrees: $3.6$ degrees to reach zero, then $2.1$ more.

Write the decimal as a fraction. It has three decimal places, so it counts thousandths, and $125$ divides both $875$ and $1000$.

$0.875 = \frac{875}{1000} = \frac{7}{8}$

So the two numbers are the same point on the line, and $-\tfrac{7}{8} = -0.875$.

Each $1 - 1$ is $0$. The fraction bar means divide, so $\dfrac{-2.5}{1 - 1}$ asks for a number that times $0$ gives $-2.5$, and no number does.

$\frac{-2.5}{1 - 1} = \frac{-2.5}{0} \;\text{is undefined}$

The others are $0$, $0$ and $-2.5$, because zero may be divided, multiplied or subtracted.

Write the decimal as a fraction: $-0.4 = -\tfrac{4}{10} = -\tfrac{2}{5}$. Its reciprocal is $-\tfrac{5}{2}$, which keeps the minus sign.

$(-0.4) \times (-2.5) = 1$

The product is $1$, so the reciprocal is $-2.5$. A reciprocal never changes the sign, because the product has to be the positive number $1$.