To find the fraction of the total number of pens that was sold, we'll break the problem down step-by-step.
1. Identify Variables:
- Let [tex]\( r \)[/tex] be the number of red pens.
- Let [tex]\( b \)[/tex] be the number of black pens.
2. Set Up Equations Based on Information Given:
- We know that [tex]\(\frac{1}{6}\)[/tex] of the red pens and [tex]\(\frac{2}{5}\)[/tex] of the black pens were sold, making the total pens sold 154:
[tex]\[
\frac{1}{6}r + \frac{2}{5}b = 154
\][/tex]
- Additionally, [tex]\(\frac{8}{11}\)[/tex] of the 154 pens sold were black:
[tex]\[
\frac{2}{5}b = \frac{8}{11} \times 154
\][/tex]
3. Determine the Number of Black Pens Sold:
- Calculate [tex]\(\frac{8}{11} \times 154\)[/tex]:
[tex]\[
\frac{8}{11} \times 154 = \frac{1232}{11} = 112
\][/tex]
- Now, [tex]\(\frac{2}{5}b = 112\)[/tex].
4. Solve for the Number of Black Pens ([tex]\( b \)[/tex]):
[tex]\[
\frac{2}{5}b = 112 \implies b = 112 \times \frac{5}{2} = 280
\][/tex]
5. Solve for the Number of Red Pens ([tex]\( r \)[/tex]):
- Substitute [tex]\( b = 280 \)[/tex] into the first equation:
[tex]\[
\frac{1}{6}r + \frac{2}{5} \times 280 = 154
\][/tex]
- Simplify the equation:
[tex]\[
\frac{1}{6}r + 112 = 154
\][/tex]
- Subtract 112 from both sides:
[tex]\[
\frac{1}{6}r = 42
\][/tex]
- Multiply by 6 to solve for [tex]\( r \)[/tex]:
[tex]\[
r = 42 \times 6 = 252
\][/tex]
6. Calculate the Total Number of Pens:
[tex]\[
\text{Total number of pens} = r + b = 252 + 280 = 532
\][/tex]
7. Determine the Fraction of the Total Pens Sold:
- The total pens sold is 154.
- The total number of pens is 532.
- Hence, the fraction of the total pens sold is:
[tex]\[
\frac{\text{Total pens sold}}{\text{Total number of pens}} = \frac{154}{532} = \frac{77}{266} \approx 0.289473684210526
\][/tex]
Therefore, the fraction of the total number of pens that was sold is approximately [tex]\(\frac{77}{266}\)[/tex] or 0.289473684210526.
