Respuesta :
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
[tex]n_r=65\%\times 560=\dfrac{65}{100}\times 560=364,[/tex]
and number of blue marbles is
[tex]n_b=560-364=196.[/tex]
Now, if 28 red marbles are replaced by blue marbles, the the new number of red and blue marbles will be
[tex]n_r^\prime=364-28=336~~~\textup{and}~~~n_b^\prime=196+28=224.[/tex]
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
[tex]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.[/tex]
Thus, 400 blue marbles need to be added to the bag.
[tex]\boxed{{\mathbf{400}}}[/tex] marbles are needed to add in the bag to maintain [tex]65\%[/tex] of all marbles are 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 It has been found that 28 red marbles has been replaced by blue marbles.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 initially 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 that means [tex]35\%[/tex] of the total marbles.
The total number of blue marbles can be evaluated 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 after the replacement can be evaluated as,
[tex]\begin{aligned}{\text{number of red marbles}} &= 364 - 28\\&= 336\\\end{aligned}[/tex]
Now blue marbles after the replacement 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 so that [tex]65\%[/tex] of all marbles are blue.
Now determine 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 and use the distributive property 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
