The number of yellow, red, and blue pens in a box is in the ratio 2:5:7. If 18 yellow pens, 12 red pens, and some blue pens are added to the box, the ratio of yellow, red, and blue pens becomes 3:6:8. Determine the number of blue pens added.

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The correct answer is 8.

Step 1: Express the initial quantities in terms of a variable.
Let the common ratio multiplier for the initial quantities be $x$.
Based on the given ratio of yellow, red, and blue pens ($2:5:7$), the initial numbers of pens are:

Initial Yellow pens = $2x$

Initial Red pens = $5x$

Initial Blue pens = $7x$

Step 2: Express the quantities after adding pens.
According to the question, 18 yellow pens, 12 red pens, and $k$ blue pens are added to the box. The new quantities become:

New Yellow pens = $2x+18$

New Red pens = $5x+12$

New Blue pens = $7x+k$

Step 3: Find the value of $x$ using the ratio of Yellow to Red pens.
The new ratio of yellow, red, and blue pens is $3:6:8$.
Using the ratio of yellow to red pens:

$\frac{2x+18}{5x+12}=\frac{3}{6}$

Simplify $\frac{3}{6}$ to $\frac{1}{2}$:

$\frac{2x+18}{5x+12}=\frac{1}{2}$

Cross-multiplying gives:

$2(2x+18)=1(5x+12)$

$4x+36=5x+12$

$5x-4x=36-12$

$x=24$

Step 4: Determine the number of blue pens added ($k$).
Substitute $x=24$ into the new quantities of Red and Blue pens:

New Red pens = $5\left(24\right)+12=120+12=132$

New Blue pens = $7\left(24\right)+k=168+k$

Now, set up the ratio of Red pens to Blue pens, which is given as $6:8$ (or $3:4$):

$\frac{132}{168+k}=\frac{6}{8}$

Simplify $\frac{6}{8}$ to $\frac{3}{4}$:

$\frac{132}{168+k}=\frac{3}{4}$

Cross-multiplying gives:

$3(168+k)=132\times 4$

$504+3k=528$

$3k=528-504$

$3k=24$

$k=8$

Thus, the number of blue pens added to the box is 8.

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The correct answer is 8.

Step 1: Express the initial quantities in terms of a variable.
Let the common ratio multiplier for the initial quantities be $x$.
Based on the given ratio of yellow, red, and blue pens ($2:5:7$), the initial numbers of pens are:

Initial Yellow pens = $2x$

Initial Red pens = $5x$

Initial Blue pens = $7x$

Step 2: Express the quantities after adding pens.
According to the problem statement, 18 yellow pens, 12 red pens, and $k$ blue pens are added to the box. The updated quantities become:

New Yellow pens = $2x+18$

New Red pens = $5x+12$

New Blue pens = $7x+k$

Step 3: Calculate the value of $x$ using the ratio of Yellow to Red pens.
The new ratio of yellow, red, and blue pens is given as $3:6:8$.
Equating the ratio of yellow pens to red pens:

$\frac{2x+18}{5x+12}=\frac{3}{6}$

Simplifying $\frac{3}{6}$ to $\frac{1}{2}$:

$\frac{2x+18}{5x+12}=\frac{1}{2}$

Cross-multiplying to solve for $x$:

$2(2x+18)=1(5x+12)$

$4x+36=5x+12$

$5x-4x=36-12$

$x=24$

Step 4: Determine the number of blue pens added ($k$).
Substituting $x=24$ into the expressions for the new quantities of Red and Blue pens:

New Red pens = $5\left(24\right)+12=120+12=132$

New Blue pens = $7\left(24\right)+k=168+k$

Setting up the ratio of Red pens to Blue pens, which is $6:8$ (or simplified to $3:4$):

$\frac{132}{168+k}=\frac{3}{4}$

Cross-multiplying to solve for $k$:

$3(168+k)=132\times 4$

$504+3k=528$

$3k=528-504$

$3k=24$

$k=8$

Therefore, the number of blue pens added to the box is 8.