Tony, Ken and Steve had some marbles. Tony had 4/5 as many marbles as Ken. Ken had 3/5 as many marbles as Steve. After Tony gave 36 marbles to Ken, the ratio of Tony's marbles to Ken's became 5:7. How many more marbles did Steve have than Ken in the end?
At the start:
Steve --- x
Ken ----- 3x/5
Tony --- (4/5)(3x/5) = 12x/25
after the give-aways
Tony --- 12x/25 - 36
Ken --- 3x/5 + 36
(12x/5 - 36) / (3x/5 + 36) = 5/7
cross-multiply
84x/5 - 252 = 15x/5 + 180
69x/5 = 432
x = 432(5/69) = not a whole number.
Find my error or else the question is bogus.
t = 4/5 k
k = 3/5 s
(t-36)/(k+36) = 5/7
solve to get
k = 720
s = 1200
t = 576
so, s-k = 540/756 1200-756 = 444
Found my error, a copy error
(12x/5 - 36) / (3x/5 + 36) = 5/7
should have been
(12x/25 - 36) / (3x/5 + 36) = 5/7
To solve this problem, we can break it down into a series of steps:
Step 1: Understand the given information.
- Tony has 4/5 as many marbles as Ken.
- Ken has 3/5 as many marbles as Steve.
- After Tony gave 36 marbles to Ken, the ratio of Tony's marbles to Ken's became 5:7.
Step 2: Assign variables to the unknown quantities.
- Let's assume that Ken originally had K marbles.
- Tony had 4/5 as many marbles as Ken, so Tony had (4/5)K marbles.
- Ken had 3/5 as many marbles as Steve, so Steve had (5/3)K marbles.
Step 3: Set up an equation based on the given information.
- After Tony gave 36 marbles to Ken, the new ratio of Tony's marbles to Ken's is 5:7.
- This means that (4/5)K - 36 : K = 5 : 7.
Step 4: Solve the equation to find the value of K.
- Cross-multiplying, we get (4/5)K - 36 = (5/7)K.
- Multiplying both sides of the equation by 35, we get 28K - 1260 = 25K.
- Simplifying, we find 3K = 1260.
- Dividing both sides of the equation by 3, we get K = 420.
Step 5: Calculate the number of marbles Steve had in the end.
- We know that Steve had (5/3)K marbles.
- Substituting the value of K, we get Steve had (5/3) * 420 = 700 marbles.
Step 6: Calculate the number of marbles Ken had in the end.
- After Tony gave 36 marbles to Ken, the new ratio of Tony's marbles to Ken's is 5:7.
- Since the new ratio is 5:7, we can set up the equation (4/5)K - 36 : K = 5 : 7.
- Substituting the value of K, we get (4/5) * 420 - 36 = 5/7 * 420.
- Calculating, we find Ken had (336 - 36) = 300 marbles.
Step 7: Calculate the difference between the number of marbles Steve and Ken had in the end.
- Steve had 700 marbles and Ken had 300 marbles.
- The difference is 700 - 300 = 400 marbles.
- Therefore, Steve had 400 more marbles than Ken in the end.