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Mathematics

Simplify :

(i) 512×(−36)\dfrac{5}{12} \times (-36)

(ii) −1718×12\dfrac{-17}{18} \times 12

(iii) −56×65\dfrac{-5}{6} \times \dfrac{6}{5}

(iv) −14×928-14 \times \dfrac{9}{28}

(v) −445×(−712)-4\dfrac{4}{5} \times \Big(-7\dfrac{1}{2}\Big)

(vi) −815×−2532\dfrac{-8}{15} \times \dfrac{-25}{32}

Rational Numbers

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Answer

(i) 512×(−36)\dfrac{5}{12} \times (-36)

Express -36 as −361\dfrac{-36}{1}

We have:

=512×−361=51×−31=5×(−3)1×1=−151=−15\phantom{=} \dfrac{5}{12} \times \dfrac{-36}{1} \\[1em] = \dfrac{5}{1} \times \dfrac{-3}{1} \\[1em] = \dfrac{5 \times (-3)}{1 \times 1} \\[1em] = \dfrac{-15}{1} = -15

Hence the answer is -15

(ii) −1718×12\dfrac{-17}{18} \times 12

Express 12 as 121\dfrac{12}{1}

We have:

=−1718×121=−173×21=−17×23×1=−343\phantom{=} \dfrac{-17}{18} \times \dfrac{12}{1} \\[1em] = \dfrac{-17}{3} \times \dfrac{2}{1} \\[1em] = \dfrac{-17 \times 2}{3 \times 1} \\[1em] = \dfrac{-34}{3}

Hence the answer is −343\dfrac{-34}{3}

(iii) −56×65\dfrac{-5}{6} \times \dfrac{6}{5}

We have:

=−56×65=−61×16=−11×11=−1×11×1=−11=−1\phantom{=} \dfrac{-5}{6} \times \dfrac{6}{5} \\[1em] = \dfrac{-6}{1} \times \dfrac{1}{6} \\[1em] = \dfrac{-1}{1} \times \dfrac{1}{1} \\[1em] = \dfrac{-1 \times 1}{1 \times 1} \\[1em] = \dfrac{-1}{1} = -1

Hence the answer is -1

(iv) −14×928-14 \times \dfrac{9}{28}

Express -14 as −141\dfrac{-14}{1}

We have:

=−141×928=−11×92=−1×91×2=−92\phantom{=} \dfrac{-14}{1} \times \dfrac{9}{28} \\[1em] = \dfrac{-1}{1} \times \dfrac{9}{2} \\[1em] = \dfrac{-1 \times 9}{1 \times 2} \\[1em] = \dfrac{-9}{2}

Hence the answer is −92\dfrac{-9}{2}

(v) −445×(−712)-4\dfrac{4}{5} \times \Big(-7\dfrac{1}{2}\Big)

We have:

=−245×−152[Converting mixed to improper fraction]=−125×−151=−121×−31=−12×(−3)1×112×31×1[product of two negative integers is positive]=361=36\phantom{=} \dfrac{-24}{5} \times \dfrac{-15}{2} \quad \text{[Converting mixed to improper fraction]} \\[1em] = \dfrac{-12}{5} \times \dfrac{-15}{1} \\[1em] = \dfrac{-12}{1} \times \dfrac{-3}{1} \\[1em] = \dfrac{-12 \times (-3)}{1 \times 1} \\[1em] \dfrac{12 \times 3}{1 \times 1} \quad \text{[product of two negative integers is positive]} \\[1em] = \dfrac{36}{1} = 36

Hence the answer is 36

(vi) −815×−2532\dfrac{-8}{15} \times \dfrac{-25}{32}

We have:

=−815×−2532=−115×−254=−13×−54=−1×(−5)3×4=1×53×4[product of two negative integers is positive]=512\phantom{=} \dfrac{-8}{15} \times \dfrac{-25}{32} \\[1em] = \dfrac{-1}{15} \times \dfrac{-25}{4} \\[1em] = \dfrac{-1}{3} \times \dfrac{-5}{4} \\[1em] = \dfrac{-1 \times (-5)}{3 \times 4} \\[1em] = \dfrac{1 \times 5}{3 \times 4} \quad \text{[product of two negative integers is positive]} \\[1em] = \dfrac{5}{12}

Hence the answer is 512\dfrac{5}{12}

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