Question

If one-eighth of a pencil is black, half of the remaining is yellow and the remaining $3\dfrac{1}{2}$ cm is blue, then what is the total length of the pencil?

Answer 1

Hint: Let x be the length of the pencil.
Then, find the part of the pencil that is black and the part of the pencil that is yellow.
Finally, to get the required answer, use the formula the length of the pencil x, can be given by the sum of the part of pencil that is black, part of the pencil that is yellow and the part of the pencil that is blue.

Complete step-by-step answer:
Let x be the length of the pencil.
Here, it is given that one-eighth of the pencil is black.
So, the portion of pencil that is black is $b = \dfrac{1}{8} \times x = \dfrac{x}{8}$ .
Thus, the remaining part of the pencil is $x - \dfrac{x}{8} = \dfrac{{8x - x}}{8} = \dfrac{{7x}}{8}$ .
Now, it is also given that half of the remaining part i.e. $\dfrac{{7x}}{8}$ part is yellow.
So, the part of the pencil that is yellow is $y = \dfrac{{\dfrac{{7x}}{8}}}{2} = \dfrac{{7x}}{{2 \times 8}} = \dfrac{{7x}}{{16}}$ .
Also, remaining $3\dfrac{1}{2}$ cm is blue i.e. $\dfrac{{\left( {3 \times 2} \right) + 1}}{2} = \dfrac{{6 + 1}}{2} = \dfrac{7}{2}$ cm.

Now, the length of the pencil x, can be given by the sum of the part of pencil that is black, part of the pencil that is yellow and the part of the pencil that is blue.
 $
  \Rightarrow x = \dfrac{x}{8} + \dfrac{{7x}}{{16}} + \dfrac{7}{2} \\
  \Rightarrow x - \dfrac{x}{8} - \dfrac{{7x}}{{16}} = \dfrac{7}{2} \\
  \Rightarrow \dfrac{{16x - 2x - 7x}}{{16}} = \dfrac{7}{2} \\
  \Rightarrow \dfrac{{7x}}{{16}} = \dfrac{7}{2} \\
  \Rightarrow x = \dfrac{7}{2} \times \dfrac{{16}}{7} \\
  \Rightarrow x = 8cm \\
 $
Thus, we get the length of the pencil as x = 8 cm.

Note: Alternate method:
Let x be the length of the pencil.
Here, it is given that one-eighth of the pencil is black.
So, the portion of pencil that is black is $b = \dfrac{1}{8} \times x = \dfrac{x}{8}$ .
Thus, the remaining part of the pencil is $x - \dfrac{x}{8} = \dfrac{{8x - x}}{8} = \dfrac{{7x}}{8}$ .
Now, it is also given that half of the remaining part i.e. $\dfrac{{7x}}{8}$ part is yellow.
So, the part of the pencil that is yellow is $y = \dfrac{{\dfrac{{7x}}{8}}}{2} = \dfrac{{7x}}{{2 \times 8}} = \dfrac{{7x}}{{16}}$ . Also, the other remaining part is also $\dfrac{{7x}}{{16}}$ .
Also, given that, remaining $3\dfrac{1}{2}$ cm is blue i.e. $\dfrac{{\left( {3 \times 2} \right) + 1}}{2} = \dfrac{{6 + 1}}{2} = \dfrac{7}{2}$ cm.
So, $\dfrac{{7x}}{{16}} = \dfrac{7}{2}$
 $
  \therefore x = \dfrac{7}{2} \times \dfrac{{16}}{7} \\
  \therefore x = 8cm \\
 $
Thus, we get the length of the pencil as \[x = 8\] cm.