27. Divisions with signs
Work out and simplify.
f) $\displaystyle \dfrac{6}{5} : \left(-\dfrac{8}{15}\right) : \left(-\dfrac{2}{3}\right)$
Two minus signs: the result is positive. Divide by multiplying by the reciprocal, and break into factors before multiplying:
$$\dfrac{6}{5} : \dfrac{8}{15} : \dfrac{2}{3} = \dfrac{6}{5} \cdot \dfrac{15}{8} \cdot \dfrac{3}{2} = \dfrac{(2 \cdot 3) \cdot (3 \cdot 5) \cdot 3}{5 \cdot 2^3 \cdot 2} = \dfrac{3^3}{2^3} = \dfrac{27}{8}$$
$\dfrac{27}{8}$
28. Multiplying fractions
Work out and simplify.
b) $\displaystyle \dfrac{18}{11} \cdot \dfrac{22}{9} \cdot \dfrac{3}{4}$
Break into factors and simplify before multiplying:
$$\dfrac{18}{11} \cdot \dfrac{22}{9} \cdot \dfrac{3}{4} = \dfrac{(2 \cdot 3^2) \cdot (2 \cdot 11) \cdot 3}{11 \cdot 3^2 \cdot 2^2} = \dfrac{3}{1} = 3$$
$3$
43. Combined operations
Work out the operations and, if possible, simplify the result.
a) $\displaystyle \dfrac{2}{7} \cdot \left[3 - \left(\dfrac{2}{5} + \dfrac{1}{2}\right)\right] - \left(-\dfrac{2}{5}\right)$
The order is the usual one: brackets from the inside out, then multiplication and division, and finally addition and subtraction.
First the inner brackets:
$$\dfrac{2}{5} + \dfrac{1}{2} = \dfrac{4}{10} + \dfrac{5}{10} = \dfrac{9}{10}$$
Then the square brackets:
$$3 - \dfrac{9}{10} = \dfrac{30}{10} - \dfrac{9}{10} = \dfrac{21}{10}$$
Now the multiplication:
$$\dfrac{2}{7} \cdot \dfrac{21}{10} = \dfrac{2 \cdot 3 \cdot 7}{7 \cdot 2 \cdot 5} = \dfrac{3}{5}$$
And finally the subtraction. Subtracting a negative is adding:
$$\dfrac{3}{5} - \left(-\dfrac{2}{5}\right) = \dfrac{3}{5} + \dfrac{2}{5} = \dfrac{5}{5} = 1$$
$1$