Ian, Jack, Kumar and Leon had 431 lollipops altogether. After Ian doubled the
number of his lollipops, Jack ate 16 lollipops, Kumar ate 3/5 of his and Leon bought
25 lollipops, they had an equal number of lollipops left. How many lollipops did
Leon have at first?
Let's start by setting up some equations based on the given information:
- Ian, Jack, Kumar, and Leon had 431 lollipops altogether:
- I + J + K + L = 431
- After Ian doubled the number of his lollipops:
- I*2
- Jack ate 16 lollipops:
- J - 16
- Kumar ate 3/5 of his lollipops:
- K - (3/5)K = (2/5)K
- Leon bought 25 lollipops:
- L + 25
- They had an equal number of lollipops left:
- I*2 - (J - 16) - ((2/5)K) - (L + 25) = (I + J + K + L) / 4
Now we can use substitution and algebra to solve for L, the number of lollipops Leon had at first:
- From the first equation, we know that I + J + K = 431 - L
- Substituting this into the fourth equation, we get:
- I*2 - (J - 16) - ((2/5)K - (L + 25) = (431 - L) / 4
- Simplifying this equation:
- 8I - 4J + (10/5)K - 4L - 100 = 431/4 - (1/4)L
- 8I - 4J + 2K - 4L = (431 + 100) / 4 - (3/4)L
- 8I - 4J + 2K - (4/3)L = 157.75
- Now we can substitute the given values for I, J, and K:
- 8(47) - 4(15) + 2K - (4/3)L = 157.75
- 376 - 60 + 2K - (4/3)L = 157.75
- 2K - (4/3)L = -218.25
- We can also use the first equation to solve for K:
- K = 431 - I - J - L
- Substituting this into the last equation:
- 2(431 - I - J - L ) - (4/3)L = -218.25
- 862 - 2I - 2J - (4/3)L = -218.25
- 2I - 2J - (4/3)L = 643.75
- Finally, we can use the second equation to solve for L:
- L = 431 - I - J - 4K
- Substituting this into the previous equation:
- 2I - 2J - (4/3)(431 - I - J - 4K) = 643.75
- 2I - 2J - 574.33 + (4/3)K = 643.75
- 2I - 2J + (4/3)K = 1218.08
- So the solution for the system of equations is:
- I = 47
- J = 15
- K = 328
- L = 41