a bag contains marbles of three different colours, red, blue and yellow. 2/5 of the marbles are red. The ratio of the number of blue marbles to the number of yellow marbles is 5:7. what fraction of the marbles in the bag is blue?
1/4
1/5
To find the fraction of the marbles in the bag that are blue, we need to determine the number of blue marbles in relation to the total number of marbles in the bag.
Let's proceed step by step:
Step 1: Understand the given information.
We know that 2/5 of the marbles are red. So, let's assume the total number of marbles in the bag is represented by the variable "x". Therefore, the number of red marbles would be (2/5) * x.
Step 2: Establish the ratio between blue and yellow marbles.
The given ratio between the number of blue marbles and the number of yellow marbles is 5:7. Let's assume the number of blue marbles is represented by the variable "b" and the number of yellow marbles is represented by the variable "y". Therefore, we can write the ratio as b:y = 5:7.
Step 3: Set up an equation.
Since we have two unknowns (b and y), we need to set up two equations.
From Step 1, we know that the number of red marbles is (2/5) * x.
From the given ratio in Step 2, we can establish another equation: b/y = 5/7.
Step 4: Solve the equations.
Using the first equation, we have:
(2/5) * x = red marbles
To solve for "red marbles," we need to find x in terms of the number of marbles. Rearranging the equation, we have:
x = (5/2) * red marbles
Next, using the second equation, we have:
b/y = 5/7
Cross-multiplying, we get:
7b = 5y
Step 5: Substitute the value of x into the equation.
Substituting x = (5/2) * red marbles into the equation 7b = 5y, we get:
7b = 5y * (5/2)
Simplifying the equation, we have:
7b = 25y/2
Multiplying both sides by 2, we get:
14b = 25y
Step 6: Calculate the fraction of blue marbles.
To determine the fraction of blue marbles, we need to consider the total number of marbles, which consists of blue, red, and yellow marbles.
The fraction of blue marbles is given by:
Fraction of blue marbles = b / (b + red marbles + yellow marbles)
Since we have the ratio of b to y (14b = 25y), we can substitute y = 14 and b = 25 in the equation.
Fraction of blue marbles = 25 / (25 + (2/5) * x + 14)
Therefore, the fraction of the marbles in the bag that is blue is 25 / (25 + (2/5) * x + 14).