In a bag, there are counters. The counters are all either red, blue, or yellow. The number of red counters, blue counters, and yellow counters are in the ratio 4:5:8. The number of yellow counters is 24 more than the number of blue counters. Work out the total number of counters in the bag.

Connecting Ratio and Difference: We are given that the number of yellow counters (Y) is 24 more than the number of blue counters (B). We can represent this difference as:

Y = B + 24

Relating Ratio and Difference with a Common Denominator: Since the ratio uses multiples of 4, let's express the difference (24) as a multiple of 4 as well. We can see that 24 is 6 times 4. So, we can rewrite the difference as:

Y = B + (6 * 4)

Expressing Each Color Count using Ratio: We know the total ratio represents all the counters in the bag. Let T be the total number of counters. Then, based on the ratio, we can express the number of each color as:

Red (R) = (4/17) * T

Blue (B) = (5/17) * T

Yellow (Y) = (8/17) * T

Substituting Difference with Ratio Expression: Now, we can substitute the expression for Y (from step 3) into this equation:

(8/17) * T = (5/17) * T + (6 * 4)

Solving for the Difference Between Blue and Yellow: Simplifying the equation:

(3/17) * T = 24

Finding the Total Number of Counters (T): To find T, we can isolate it by multiplying both sides by the reciprocal of (3/17):

T = 24 * (17/3)

T = 136

There are a total of 136 counters in the bag