a pouch contains 97 green beads 72 red beads and 7 yellow beads. how many blue beads should be added so that the probability of drawing a red bead is 3/10 ?
To find out how many blue beads need to be added, let's assume the total number of blue beads added to the pouch is denoted by 'x'.
First, let's calculate the total number of beads in the pouch after adding the blue beads:
Total number of green beads = 97
Total number of red beads = 72 + x
Total number of yellow beads = 7
Total number of blue beads = x
Total number of beads = total green beads + total red beads + total yellow beads + total blue beads
Total number of beads = 97 + (72 + x) + 7 + x
The probability of drawing a red bead is given by the ratio of the number of red beads to the total number of beads in the pouch, which should be equal to 3/10:
Probability of drawing a red bead = (number of red beads) / (total number of beads)
3/10 = (72 + x) / (97 + (72 + x) + 7 + x)
Now, let's solve this equation to find the value of 'x':
3/10 = (72 + x) / (176 + x)
Cross multiplying:
3(176 + x) = 10(72 + x)
528 + 3x = 720 + 10x
Rearranging terms:
10x - 3x = 720 - 528
7x =192
Dividing both sides by 7:
x = 192/7
x ≈ 27.43
Since we cannot have a fraction of a bead, we need to round up the result to the nearest whole number. Thus, we should add 28 blue beads to the pouch in order to have a probability of drawing a red bead of 3/10.
To find out how many blue beads should be added, we need to consider the probability of drawing a red bead after adding the blue beads to the pouch.
Let's start by calculating the total number of beads in the pouch before adding any blue beads:
Total number of beads = number of green beads + number of red beads + number of yellow beads
= 97 + 72 + 7
= 176
Now, let's say we add 'x' blue beads to the pouch. After adding blue beads, the total number of beads becomes:
Total number of beads = number of green beads + number of red beads + number of yellow beads + number of blue beads
= 97 + 72 + 7 + x
Since we want the probability of drawing a red bead to be 3/10, we can set up the following equation:
Number of red beads / Total number of beads = 3/10
Adding blue beads does not change the number of red beads and yellow beads, but it increases the total number of beads. Substituting the values into the equation, we get:
72 / (176 + x) = 3/10
To solve this equation, we can cross-multiply:
72 * 10 = 3 * (176 + x)
720 = 528 + 3x
Rearranging the equation, we find:
3x = 720 - 528
3x = 192
x = 192/3
x = 64
Therefore, we need to add 64 blue beads to the pouch so that the probability of drawing a red bead is 3/10.