QuestionIn a jar, $\frac{2}{5}$ are blue, $\frac{1}{3}$ are red, and the rest (126) are green and yellow. Green = $\frac{3}{4}$ yellow. Find yellow beads.

Studdy Solution

STEP 1

Assumptions1. The fraction of blue beads is $\frac{}{5}$ . The fraction of red beads is $\frac{1}{3}$
3. The total number of red, green, and yellow beads is1264. The number of green beads is $\frac{3}{4}$ of the number of yellow beads5. The rest of the beads are either green or yellow

STEP 2

First, we need to find the fraction of the beads that are either green or yellow. We can do this by subtracting the fractions of blue and red beads from1.
$Fraction_{green + yellow} =1 - Fraction_{blue} - Fraction_{red}$

STEP 3

Now, plug in the given values for the fractions of blue and red beads to calculate the fraction of green and yellow beads.
$Fraction_{green + yellow} =1 - \frac{2}{5} - \frac{1}{3}$

STEP 4

To subtract fractions, we need to find a common denominator. The least common multiple of and3 is15, so we will convert both fractions to have this denominator.
$\frac{2}{} = \frac{6}{15}, \quad \frac{1}{3} = \frac{}{15}$ $Fraction_{green + yellow} =1 - \frac{6}{15} - \frac{}{15}$

STEP 5

Subtract the fractions to find the fraction of green and yellow beads.
$Fraction_{green + yellow} =1 - \frac{}{15} - \frac{5}{15} = \frac{15}{15} - \frac{}{15} - \frac{5}{15} = \frac{4}{15}$

STEP 6

Now that we have the fraction of green and yellow beads, we can find the total number of green and yellow beads by multiplying this fraction by the total number of red, green, and yellow beads.
$Number_{green + yellow} = Fraction_{green + yellow} \times Total_{red + green + yellow}$

STEP 7

Plug in the values for the fraction of green and yellow beads and the total number of red, green, and yellow beads to calculate the number of green and yellow beads.
$Number_{green + yellow} = \frac{4}{15} \times126$

STEP 8

Calculate the number of green and yellow beads.
$Number_{green + yellow} = \frac{4}{15} \times126 =33.6$

STEP 9

Since the number of beads cannot be a fraction, we round this up to the nearest whole number. There are34 green and yellow beads.

STEP 10

We know that the number of green beads is $\frac{3}{4}$ of the number of yellow beads. We can write this as an equation and solve for the number of yellow beads.
$Number_{green} = \frac{3}{4} \times Number_{yellow}$

STEP 11

We also know that the total number of green and yellow beads is the sum of the number of green beads and the number of yellow beads.
$Number_{green + yellow} = Number_{green} + Number_{yellow}$

STEP 12

Substitute the equation from10 into the equation from11.
$Number_{green + yellow} = \frac{}{4} \times Number_{yellow} + Number_{yellow}$

STEP 13

implify the equation.
$Number_{green + yellow} = \frac{7}{} \times Number_{yellow}$

STEP 14

olve for the number of yellow beads.
$Number_{yellow} = \frac{4}{7} \times Number_{green + yellow}$

STEP 15

Plug in the value for the number of green and yellow beads to calculate the number of yellow beads.
$Number_{yellow} = \frac{4}{7} \times34$

STEP 16

Calculate the number of yellow beads.
$Number_{yellow} = \frac{4}{} \times34 =19.43$

STEP 17

Since the number of beads cannot be a fraction, we round this up to the nearest whole number. There are20 yellow beads.

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