There were some black and red pens in a shop. [tex]\frac{1}{6}[/tex] of the red pens and [tex]\frac{2}{5}[/tex] of the black pens were sold. A total of 154 pens were sold, and out of those, [tex]\frac{8}{11}[/tex] were black. What fraction of the total number of black and red pens in the shop was sold?



Answer :

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].