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?

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Answer:

400 blue marbles.

Step-by-step explanation:

We have been given that of 560 marbles in a bag, 65% are red and the rest are blue.

[tex]\text{Number of red marbles initially}=\frac{65}{100}\times 560[/tex]

[tex]\text{Number of red marbles initially}=0.65\times 560[/tex]

[tex]\text{Number of red marbles initially}=364[/tex]

[tex]\text{Number of blue marbles initially}=560-364[/tex]

[tex]\text{Number of blue marbles initially}=196[/tex]

Then, after replacement:

[tex]\text{Red marbles after replacement}=364-28=336[/tex]

[tex]\text{Blue marbles after replacement}=196+28=224[/tex]

Let x blue marbles be added. So we can set up an equation as:

[tex](\frac{224+x}{560+x})*100=65[/tex]

[tex]\frac{224+x}{560+x}=\frac{65}{100}[/tex]

Upon cross multiplying our equation we will get,

[tex](224+x)100=65(560+x)[/tex]

Using distributive property [tex]a(b+c)=a*b+a*c[/tex]

[tex]22400+100x=36400+65x[/tex]  

[tex]22400-22400+100x-65x=36400-22400+65x-65x[/tex]  

[tex]100x-65x=36400-22400[/tex]  

[tex]35x=14000[/tex]  

[tex]\frac{35x}{35}=\frac{14000}{35}[/tex]  

[tex]x=\frac{14000}{35}[/tex]  

[tex]x=400[/tex]  

Therefore, 400 blue marbles needed to be added to the bag so that the 65% of all marbles are blue.

[tex]\boxed{{\mathbf{400}}}[/tex] marbles are required to add in the bag so that [tex]65\%[/tex] of all marbles area blue.

Further explanation:

Given:

There are total 560 marbles in bag in which [tex]65\%[/tex] 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 It has been found that 28 red marbles has been replaced by blue marbles.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 [tex]65\%[/tex] are red marbles and the total number of marbles are 560.

The total number of red marbles can be calculated as,

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

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

Step 2:

It is given that rest of the marbles are blue.

The total number of blue marbles can be calculated as,

[tex]\begin{aligned}{\text{number of blue marbles}} &= 560 - 364 \\&= 196 \\\end{aligned}[/tex]  

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 can be calculated as,

[tex]\begin{aligned}{\text{number of blue marbles}} &= 560 - 364 \\&= 196 \\\end{aligned}[/tex]  

Now blue marbles can be calculated as,

[tex]\begin{aligned}{\text{number of blue marbles}} &= 196 + 28 \\&= 224 \\\end{aligned}[/tex]  

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 [tex]65\%[/tex] blue marbles.

Consider [tex]x[/tex] as the number of blue marbled added in the bag to maintain [tex]65\%[/tex]

Now make an equation for the percent of blue marbles.

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

Step 5:

Now cross multiply the resultant equation to obtain the value of [tex]x[/tex].

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

Further calculation for the value of [tex]x[/tex] can be calculated as,

[tex]\begin{aligned}x &= \frac{{14000}}{{35}} \hfill \\x &= 400 \hfill \\\end{aligned}[/tex]  

Therefore, 400 marbles are required to add in the bag so that [tex]65\%[/tex] 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