To solve this problem, we start by defining the variables and using the given information to set up the appropriate equations.
1. Define Variables:
- Let [tex]\( R \)[/tex] be the total number of red pens in the shop.
- Let [tex]\( B \)[/tex] be the total number of black pens in the shop.
2. Given Information:
- The total number of pens sold is 154.
- [tex]\(\frac{8}{11}\)[/tex] of the pens sold are black.
- [tex]\(\frac{1}{6}\)[/tex] of the red pens were sold.
- [tex]\(\frac{2}{5}\)[/tex] of the black pens were sold.
3. Calculate the Number of Black Pens Sold:
Since [tex]\(\frac{8}{11}\)[/tex] of the 154 pens sold are black:
[tex]\[
\text{Number of black pens sold} = \frac{8}{11} \times 154 = 112
\][/tex]
4. Calculate the Number of Red Pens Sold:
The remaining pens sold are red:
[tex]\[
\text{Number of red pens sold} = 154 - 112 = 42
\][/tex]
5. Set Up Equations for Red and Black Pens Using Given Ratios:
- According to the problem, [tex]\(\frac{1}{6}\)[/tex] of the total red pens is sold, which is 42:
[tex]\[
\frac{1}{6} R = 42 \implies R = 42 \times 6 = 252
\][/tex]
- According to the problem, [tex]\(\frac{2}{5}\)[/tex] of the total black pens is sold, which is 112:
[tex]\[
\frac{2}{5} B = 112 \implies B = 112 \times \frac{5}{2} = 280
\][/tex]
6. Calculate the Total Number of Pens in the Shop:
The total number of pens in the shop is the sum of red and black pens:
[tex]\[
\text{Total number of pens} = R + B = 252 + 280 = 532
\][/tex]
7. Calculate the Fraction of the Total Number of Pens Sold:
The fraction of pens sold is the total pens sold divided by the total number of pens in the shop:
[tex]\[
\text{Fraction of pens sold} = \frac{154}{532} \approx 0.29
\][/tex]
Therefore, the fraction of the total number of black and red pens in the shop that was sold is approximately [tex]\( \frac{154}{532} \)[/tex], or roughly [tex]\( 0.29 \)[/tex].
