There are a total of 700 blue, yellow, and red balls in a machine.
The ratio of blue balls to the total number of balls is 2:7.
The ratio of yellow balls to blue balls is 7:4.
The ratio of blue balls to red is 4:3.
With the given information calculate the number of blue balls, yellow balls and red balls.
Blue balls = 2:7 = 200/700 = 200
Blue to red = 4:3 = 200:x, red = 150
total - red - blue = yellow
To calculate the number of blue balls, yellow balls, and red balls, we'll first assign variables to represent their quantities. Let's use B for blue balls, Y for yellow balls, and R for red balls.
According to the given information, the ratio of blue balls to the total number of balls is 2:7. This means that out of the total number of balls, there are 2 parts that are blue and 7 parts in total. We can express this as the equation:
B / (B + Y + R) = 2 / 7
Next, we're told that the ratio of yellow balls to blue balls is 7:4. This means that for every 7 parts blue balls, there are 4 parts yellow balls. We can express this as the equation:
Y / B = 7 / 4
Finally, the ratio of blue balls to red balls is 4:3. This means that for every 4 parts blue balls, there are 3 parts red balls. We can express this as the equation:
B / R = 4 / 3
Now we can solve these equations simultaneously to find the values of B, Y, and R.
Step 1: Solve the second equation for Y in terms of B:
Y = (7/4)B
Step 2: Substitute the value of Y in terms of B into the first equation:
B / (B + (7/4)B + R) = 2 / 7
Step 3: Simplify the equation from step 2:
B / (B(1 + 7/4) + R) = 2 / 7
B / (B(11/4) + R) = 2 / 7
Step 4: Cross-multiply and simplify the equation from step 3:
7B = 2(B(11/4) + R)
7B = (22/4)B + 2R
Step 5: Multiply through by 4 to eliminate the fractions:
28B = 11B + 8R
Step 6: Subtract 11B from both sides to isolate R:
17B = 8R
Step 7: Since the ratio of B to R is 4:3, we can rewrite the equation as:
17B = 8(4/3)B
Step 8: Multiply through by 3 to eliminate the fractions:
51B = 32B
Step 9: Divide both sides by B to solve for B:
51 = 32
Step 10: Since we have reached a contradiction (51 is not equal to 32), it means that there is no consistent solution for this system of equations.
Therefore, based on the given information, it is not possible to determine the exact number of blue balls, yellow balls, and red balls.