Respuesta :
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 It has been found that 28 red marbles has been replaced by blue marbles.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.
Learn more:
- Learn more about the solution of the linear equation https://brainly.com/question/11744034
- Learn more about the problem of the linear equation https://brainly.com/question/2550559
- Learn more about equations of the area and perimeter of the rectangle https://brainly.com/question/1511313
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