The sum of beads owned by John, Bob, and Dale is 273. The ratio of John's beads and Dale's is 8:5, and Dale's beads are 10/13 times Bob's. How many more beads does John own than Dale?
John=112 beads
Bob=91 beads
Dale=70
So john have 42 bead more than Dale
John's number --- 8x
Dale's number --- 5x
Bob's number --- y
but Dale's = (10/13)Bob's
5x = (10/13)Bob
Bob = 5x(13/10) = 13x/2
finally:
8x + 5x + 13x/2 = 273
16x + 10x + 13x = 546
39x = 546
x = 14
John has 112
Dale has 70
Bob has 91
So John has 42 more than Dale
To find out how many more beads John owns than Dale, we need to determine the number of beads owned by each of them.
Let's start by assigning variables to the number of beads owned by John, Bob, and Dale. Let J represent John's number of beads, B represent Bob's number of beads, and D represent Dale's number of beads.
We are given that the sum of beads owned by all three is 273, so we can write the equation J + B + D = 273.
We are also given that the ratio of John's beads to Dale's beads is 8:5. This can be written as J : D = 8 : 5. Multiplying the ratio by a common factor, we get 8x : 5x, where x is a positive constant.
Lastly, we are told that Dale's beads are 10/13 times Bob's beads. This can be written as D = (10/13)B.
Now, let's solve this system of equations to find the values of J, B, and D.
Using the given ratio, we can rewrite the equation as J = (8/5)D.
Substituting D in terms of B from the third piece of given information, we get D = (10/13)B.
Now substitute D with (10/13)B in the equation J + B + D = 273:
(8/5)D + B + (10/13)B = 273
Multiplying through to remove the fractions:
8(10/13)B + 5B + 10(10/13)B = 273
Simplifying:
80/13B + 5B + 100/13B = 273
Combining like terms:
(80/13 + 100/13)B + 5B = 273
180/13B + 5B = 273
Multiplying through by 13 to get rid of the denominator:
180B + 65B = 13 * 273
245B = 3549
B = 3549/245
B ≈ 14.48
Since B represents Bob's number of beads, it should be a whole number. Therefore, we should round it to the nearest whole number:
B ≈ 14
Now we can find D using D = (10/13)B:
D = (10/13) * 14
D ≈ 10.77
Again, since D represents the number of beads and it should be a whole number, we will round it to the nearest whole number:
D ≈ 11
Let's find J using J = (8/5)D:
J = (8/5) * 11
J ≈ 17.6
Rounding it to the nearest whole number, we get:
J ≈ 18
Now we can find the difference between the number of beads owned by John and Dale:
Difference = J - D = 18 - 11 = 7
Therefore, John owns 7 more beads than Dale.