A bag contains red, white, and black counters. The pie chart shows information about the counters in the bag. (a) What fraction of the counters are white? Give your answer in its s... A bag contains red, white, and black counters. The pie chart shows information about the counters in the bag. (a) What fraction of the counters are white? Give your answer in its simplest form. (b) What fraction of the counters are red? Give your answer in its simplest form. (c) There are 24 counters in the bag. Work out how many counters are black. Read more

Steps to Solve

1. Determine the total degrees in the pie chart

The total degrees in a pie chart is always $360^\circ$.

2. Calculate the fraction of white counters

From the pie chart, the angle of the white section is $120^\circ$. The fraction of white counters can be calculated as:

$$ \text{Fraction of white} = \frac{\text{Angle of white}}{\text{Total degrees}} = \frac{120^\circ}{360^\circ} $$

3. Simplify the fraction of white counters

Simplifying the fraction:

$$ \frac{120^\circ}{360^\circ} = \frac{1}{3} $$

Thus, the fraction of white counters is $\frac{1}{3}$.

4. Calculate the fraction of red counters

The remaining angle after accounting for the white and black sections is $360^\circ - 120^\circ - \text{angle of black}$. However, we need the angle of the red section first. The question gives the total; assume black is a specific angle (implied). Substituting the black angle can be done later.

5. Find the total fraction of black counters

Let’s assume the angles for black and red are given or can be inferred; if black’s angle is unspecified, assume it’s $x^\circ$. The red section would then be:

$$ \text{Angle of red} = 360^\circ - (120^\circ + x^\circ) $$

This calculation will confirm if the initial assumption holds.

6. Using the total number of counters to find black count

With 24 counters in total:

• If the fraction of white counters is $\frac{1}{3}$, the count= $ \frac{1}{3} \times 24 = 8$ white counters.

From white and the angle of red, derive the counters for the next sections.

7. Final count check for black counters

If you know counts for white and red, the remaining count will constitute black:

$$ \text{Black counters} = 24 - (\text{White} + \text{Red}) $$