# (a)(i) Provide the number in the box $\boxed{}$, such that $\frac{2}{3}\times\boxed{}=\frac{10}{30}$(ii) The simplest form of the number obtained in $\boxed{}$ is _____.(b) (i) Provide the number in the box $\boxed{}$, such that $\frac{3}{5}\times\boxed{}=\frac{24}{75}$(ii) The simplest form of the number obtained in $\boxed{}$ is _____.

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To do:

We have to find the missing numbers.

Solution:

(a)

(i) $\frac{2}{3}\times$________$=\frac{10}{30}$

Let the missing number be $\frac{x}{y}$

This implies,

$\frac{2}{3}\times \frac{x}{y}=\frac{10}{30}$

$\frac{2\times x}{3\times y}=\frac{10}{30}$

$2x=10$ and $3y=30$

$x=\frac{10}{2}$ and $y=\frac{30}{3}$

$x=5$ and $y=10$

Therefore, $\frac{5}{10}$ is the number in the box, such that $\frac{2}{3}\times\frac{5}{10}=\frac{10}{30}$.

(ii) The simplest form of $\frac{5}{10}$ is

$\frac{5}{10}=\frac{5\times1}{5\times2}$

$=\frac{1}{2}$

(b)

(i) $\frac{3}{5}\times$______$=\frac{24}{75}$

Let the missing number be $\frac{x}{y}$

This implies,

$\frac{3}{5}\times \frac{x}{y}=\frac{24}{75}$

$\frac{3\times x}{5\times y}=\frac{24}{75}$

$3x=24$ and $5y=75$

$x=\frac{24}{3}$ and $y=\frac{75}{5}$

$x=8$ and $y=15$

Therefore, $\frac{8}{15}$ is the number in the box, such that $\frac{3}{5}\times\frac{8}{15}=\frac{24}{75}$.

(ii) As $\frac{8}{15}$ can't be simplified further since the GCF of the numerator and the denominator is $1$.

Therefore, its simplest form is $\frac{8}{15}$ itself.

Updated on 10-Oct-2022 13:32:36