Question

Of 560 marbles in a bag, 65% are red and the rest are blue. After 28 red marbles are replaced with blue ones, how many blue marbles need to be added to the bag so that the 65% of all marbles are blue?

Answer 1

$\boxed{{\mathbf{400}}}$ marbles are needed to add in the bag to maintain $65\%$ of all marbles are blue.

Further explanation:

Given  

There are total 560 marbles in bag in which $65\%$ are red and the rest are blue. It has been found that 28 red marbles has been replaced by blue marbles.

Step by step explanation:

Step 1:

It is given that $65\%$ are red marbles and the total number of marbles are 560.

The total number of red marbles initially can be calculated as,

$\begin{aligned}{\text{number of red marbles}} &= \frac{{65}}{{100}} \times 560 = 0.65 \times 560\\&= 364\\\end{aligned}$

Therefore, the total number of red marbles initially are 364.

Step 2:

It is given that rest of the marbles are blue that means $35\%$ of the total marbles.

The total number of blue marbles can be evaluated as,

$\begin{aligned}{\text{number of blue marbles}} &= 560 - 364\\&= 196 \\\end{aligned}$  

Therefore, the total number of blue marbles initially are 364.

Step 3:

Now it is given that the 28 red marbles are replaced with blue marbles.

Now red marbles after the replacement can be evaluated as,

$\begin{aligned}{\text{number of red marbles}} &= 364 - 28\\&= 336\\\end{aligned}$  

Now blue marbles after the replacement can be calculated as,

$\begin{aligned}{\text{number of blue marbles}} &= 196 + 28 \\&= 224 \\\end{aligned}$  

Therefore, the number of red marbles are 336 and the number of blue marbles are 224.

Step 4:

Now find the number of blue marbles added in the bag to maintain $65\%$ blue marbles.

Consider $x$ as the number of blue marbled added in the bag so that $65\%$ of all marbles are blue.

Now determine an equation for the percent of blue marbles.

$\begin{aligned}\left( {\frac{{224 + x}}{{560 + x}}} \right) \times 100 &= 65 \\ \frac{{224 + x}}{{560 + x}} &= \frac{{65}}{{100}} \\\end{aligned}$  

Step 5:

Now cross multiply the resultant equation and use the distributive property to obtain the value of $x$.

$\begin{aligned}\left( {224 + x} \right)100 &= 65\left( {560 + x} \right) \\22400 + 100x &= 36400 + 65x \\100x - 65x &= 36400 - 22400 \\35x &= 14000 \\\end{aligned}$  

Further calculation for the value of $x$ can be calculated as,

$\begin{aligned}x&= \frac{{14000}}{{35}} \hfill \\x &= 400 \hfill \\\end{aligned}$  

Therefore, 400 marbles are required to add in the bag so that $65\%$ of all marbles area blue.

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Answer details:

Grade: High school

Subject: Mathematics

Chapter: Permutation

Keywords: Blue marbles, red marbles, bag, cross multiply, added, replaced, total marbles, initially, replacement, distributive property, numbers

Answer 2

Answer:  The answer is 400  blue marbles.

Step-by-step explanation: Given that there are 560 marbles in a bag, out of which 65% are red and rest are blue.

So, number of red marbles is

$n_r=65\%\times 560=\dfrac{65}{100}\times 560=364,$

and number of blue marbles is

$n_b=560-364=196.$

Now, if 28 red marbles are replaced by blue marbles, the the new number of red and blue marbles will be

$n_r^\prime=364-28=336~~~\textup{and}~~~n_b^\prime=196+28=224.$

Now, to get 65% of the marbles blue, we need to add some more blue marbles to the bag. Let 'x' number of blue marbles are added to the bag, then

$65\%\times (560+x)=224+x\\\\\Rightarrow \dfrac{65}{100}\times (560+x)=224+x\\\\\Rightarrow 13(560+x)=20(224+x)\\\\\Rightarrow 7280+13x=4480+20x\\\\\Rightarrow 7x=2800\\\\\Rightarrow x=400.$

Thus, 400 blue marbles need to be added to the bag.