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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?

Sagot :

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.
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