Operations with Signed Rational Numbers: Additional Practice (Set 1)

The decimal $0.125$ is $\tfrac{125}{1000} = \tfrac{1}{8}$. Both numbers are negative, so add the sizes and keep the sign.

$\frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2}$

So the sum is $-\tfrac{1}{2}$.

Subtracting $-\tfrac{3}{4}$ is adding $\tfrac{3}{4} = \tfrac{9}{12}$. The signs then differ, and $\tfrac{9}{12}$ is the larger size.

$-\frac{5}{12} + \frac{9}{12} = \frac{4}{12} = \frac{1}{3}$

The signs match, so the quotient is positive. Write $0.75 = \tfrac{3}{4}$ and multiply by the reciprocal of $\tfrac{1}{8}$.

$\frac{3}{4} \times 8 = \frac{24}{4} = 6$

The choice $\tfrac{3}{32}$ multiplies by $\tfrac{1}{8}$ instead of dividing.

Both factors are negative, so the product is positive. Write the mixed numbers as fractions: $2\tfrac{1}{3} = \tfrac{7}{3}$ and $1\tfrac{2}{7} = \tfrac{9}{7}$.

$\frac{7}{3} \times \frac{9}{7} = \frac{63}{21} = 3$

The choice $2\tfrac{2}{21}$ multiplies the whole parts and the fraction parts separately, which is not how a product of mixed numbers works.

The positive $0.1$ is greatest. Among the negatives, the one closest to zero is greatest, so compare the sizes $\tfrac{1}{9}$, $0.12$ and $\tfrac{1}{8}$. Since $0.12 = \tfrac{3}{25}$, compare it with $\tfrac{1}{9}$ over $225$ths.

$\frac{1}{9} = \frac{25}{225} < \frac{27}{225} = \frac{3}{25}$

Also $0.12 < 0.125 = \tfrac{1}{8}$. So the sizes run $\tfrac{1}{9} < 0.12 < \tfrac{1}{8}$, and the order is $0.1, -\tfrac{1}{9}, -0.12, -\tfrac{1}{8}$.

Rising adds, so the new position is $-42.5 + 18.75$. The signs differ, so subtract the smaller size from the larger, lining up the points.

$42.50 - 18.75 = 23.75$

The number $-42.5$ is farther from zero, so the submarine is at $-23.75$ meters, still below the surface.

The distance between them is $0.5 - (-1.5) = 2$, so halfway is $1$ unit from each end.

$-1.5 + 1 = -0.5$

Check from the other end: $0.5 - 1 = -0.5$ as well.

Three negative factors make an odd count, so the product is negative. For the size, multiply as whole numbers and count three decimal places.

$2 \times 3 \times 5 = 30, \quad \text{so} \quad 0.2 \times 0.3 \times 0.5 = 0.030$

So the product is $-0.03$.

A quotient of two negatives is positive. Its value is $\tfrac{2}{3} \times \tfrac{5}{4}$.

$\left(-\frac{2}{3}\right) \div \left(-\frac{4}{5}\right) = \frac{10}{12} = \frac{5}{6}$

The product of different signs is negative. The difference is $-\tfrac{4}{6} + \tfrac{3}{6} = -\tfrac{1}{6}$, also negative. A product with a zero factor is $0$, which is neither positive nor negative.

The opposite of $-1\tfrac{1}{4}$ is $1\tfrac{1}{4} = 1.25$, so the problem is $1.25 + (-2.5)$. The signs differ.

$2.50 - 1.25 = 1.25$

The number $-2.5$ is farther from zero, so the answer is $-1.25$.

Start with the fraction. In $\dfrac{-3}{-4}$ the numerator and denominator have matching signs, so it is positive.

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

The minus sign in front asks for the opposite of that, which is $-\tfrac{3}{4}$.

The number needed is the result minus the starting value. Subtracting $-2.75$ is adding $2.75$.

$-1.5 - (-2.75) = -1.5 + 2.75 = 1.25$

Check: $-2.75 + 1.25 = -1.5$. The sum moved right, toward zero, so the number added had to be positive.