Question

Fill in the following blanks:
\[\dfrac{{15}}{{18}} = \dfrac{{\boxed{}}}{6} = \dfrac{{10}}{{\boxed{}}} = \dfrac{{\boxed{}}}{{30}}\] [Are these equivalent ratios?]

Answer 1

Hint:We are given with some ratios with blanks and we need to find the values that would fill the blanks. Find the values of each blank separately, start from the first and second ratio to find the value of first blank and then go for second and third ratios to find the value of second blank and so on.

We are also asked whether the ratios are equivalent for this recall the definition of equivalent ratios.

Complete step by step solution:
Given, \[\dfrac{{15}}{{18}} = \dfrac{{\boxed{}}}{6} = \dfrac{{10}}{{\boxed{}}} = \dfrac{{\boxed{}}}{{30}}\]

Let us name the blank boxes for simplicity,
\[\dfrac{{15}}{{18}} = \dfrac{x}{6} = \dfrac{{10}}{y} = \dfrac{z}{{30}}\] (i)

Now, we will find the values of \[x\], \[y\] and \[z\] one by one .
To find the value of \[x\], we take the first and second ratios , that is \[\dfrac{{15}}{{18}} = \dfrac{x}{6}\]
\[\dfrac{{15}}{{18}} = \dfrac{x}{6}\]
\[ \Rightarrow x = \dfrac{{15}}{{18}} \times 6\]
\[ \Rightarrow x = \dfrac{{15}}{3}\]
\[ \Rightarrow x = 5\]

To find the value of \[y\], we take the second and third ratios, that is \[\dfrac{x}{6} = \dfrac{{10}}{y}\]
\[\dfrac{x}{6} = \dfrac{{10}}{y}\]

Putting the value of \[x\] in the above equation we get,
\[\dfrac{5}{6} = \dfrac{{10}}{y}\]
\[ \Rightarrow y = 10 \times \dfrac{6}{5}\]
\[ \Rightarrow y = 2 \times 6\]
\[ \Rightarrow y = 12\]

To find the value of \[z\], we take the third and fourth ratios, that is \[\dfrac{{10}}{y} = \dfrac{z}{{30}}\]
\[\dfrac{{10}}{y} = \dfrac{z}{{30}}\]
Putting the value of \[y\] in the above equation we get,
\[\dfrac{{10}}{{12}} = \dfrac{z}{{30}}\]
\[ \Rightarrow z = \dfrac{{10}}{{12}} \times 30\]
\[ \Rightarrow z = \dfrac{{10}}{2} \times 5\]
\[ \Rightarrow z = 5 \times 5\]
\[ \Rightarrow z = 25\]

Therefore the values of \[x\], \[y\] and \[z\] are \[5\], \[12\] and \[25\] respectively.
Putting these values in equation (i), we get
\[\dfrac{{15}}{{18}} = \dfrac{5}{6} = \dfrac{{10}}{{12}} = \dfrac{{25}}{{30}}\]

We are also asked whether the ratios are equivalent. Equivalent ratios means the ratios have the same value. To find whether the ratios are equivalent we evaluate each ratio and check their values.

The first ratio \[\dfrac{{15}}{{18}}\]
\[\dfrac{{15}}{{18}} = \dfrac{{5 \times 3}}{{6 \times 3}} = \dfrac{5}{6}\] (iii)

The second ratio \[\dfrac{5}{6}\] is already in simplest form.

The third ratio \[\dfrac{{10}}{{12}}\]
\[\dfrac{{10}}{{12}} = \dfrac{{5 \times 2}}{{6 \times 2}} = \dfrac{5}{6}\] (iv)

The fourth ratio \[\dfrac{{25}}{{30}}\]
\[\dfrac{{25}}{{30}} = \dfrac{{5 \times 5}}{{6 \times 5}} = \dfrac{5}{6}\] (v)

From equations (iii), (iv) and (v) we observe that the values of all ratios is \[\dfrac{5}{6}\]. Therefore, as the values of all ratios are the same, they are equivalent ratios.

Note:For such types of questions, all find the blanks one by one. Start from the end where both numerator and denominator are present like here the left most ratio was given so we started from the left and if in a question the rightmost ratio is given and leftmost ratio is not given completely then start from the right side.