In January, Alison, Bryan and Carly spent a total of $1510. In February, Alison increased her spending by $140. Bryan doubled his spending and Carly reduced her spending by $150. The amount of money they each spent then became the same. If Alison saved $1850 in January, find the total amount of money she had in January at first.
If Alison saved $1850 in January, the total amount of money she used in January is $1510+$1850 = $<<1850+1510=3360>>3360
In February, Alison spent $140 more than she spent in January, meaning she spent $1510+$140 = $<<1510+140=1650>>1650 in February.
Bryan spent twice as much money in February as in January, a total of $1510*2 = $3020
Carly spent $150 less in February than in January, meaning she spent $1510-$150 = $1360 in February.
Since the amount of money they each used in February was equal, and Carly used $1360, Alison used $1650 and Bryan used $3020, the total amount of money they used in February is $1360+$1650+$3020 = <<1360+1650+3020=6030>>6030
Together, in January and February, Alison used $1650+$3360 = $<<1650+3360=5010>>5010.
The total amount of money she used in January and February is $6030, meaning Alison had $$6030-$5010 = $<<6030-5010=1020>>1020 in January. Answer: \boxed{1020}.
Let's represent the amount of money Alison, Bryan, and Carly spent in January as A, B, and C, respectively.
In January, Alison, Bryan, and Carly spent a total of $1510, so we can write the equation:
A + B + C = 1510 ----(equation 1)
In February, Alison increased her spending by $140, so her new spending is A + 140.
Bryan doubled his spending, so his new spending is 2B.
Carly reduced her spending by $150, so her new spending is C - 150.
After these changes, the amount of money they each spent became the same, so we can write the equation:
A + 140 = 2B = C - 150 ----(equation 2)
If Alison saved $1850 in January, we can write the equation:
A + 1850 = Total amount of money Alison had in January ----(equation 3)
To solve these equations, we can substitute equation 2 into equation 1 and also substitute equation 3 into equation 1.
Substituting equation 2 into equation 1, we get:
(A + 140) + 2B + (C - 150) = 1510
A + 140 + 2B + C - 150 = 1510
A + 2B + C + 140 - 150 = 1510
A + 2B + C - 10 = 1510
A + 2B + C = 1520 ----(equation 4)
Substituting equation 3 into equation 1, we get:
(1850) + B + C = 1510
B + C = 1510 - 1850
B + C = -340 ----(equation 5)
Now we have two equations with two unknowns:
A + 2B + C = 1520 ----(equation 4)
B + C = -340 ----(equation 5)
We can solve these equations by substitution or elimination.
Let's solve by elimination. Multiply equation 5 by -1, we get:
-B - C = 340 ----(equation 6)
Add equation 4 and equation 6 together, we have:
(A + 2B + C) + (-B - C) = 1520 + 340
A + B = 1860
Since the amount of money they each spent became the same, A + B + C = 1510 (from equation 1).
Subtracting equation A + B = 1860 from A + B + C = 1510, we have:
A + B + C - (A + B) = 1510 - 1860
C - B = -350 ----(equation 7)
From equation 5, we have: B + C = -340
Subtracting equation 7 from equation 5, we get:
(B + C) - (C - B) = -340 - (-350)
B + C - C + B = -340 + 350
2B = 10
B = 10/2
B = 5
Substituting B = 5 into equation 5, we have:
5 + C = -340
C = -340 - 5
C = -345
Now, substituting B = 5 and C = -345 into equation 1, we have:
A + 5 - 345 = 1510
A - 340 = 1510
A = 1510 + 340
A = 1850
Therefore, the total amount of money Alison had in January at first was $1850.
To solve this problem, let's assign variables to the amounts of money each person spent in January. Let A represent Alison's spending, B represent Bryan's spending, and C represent Carly's spending in January.
We are given the equation A + B + C = 1510, which represents the total amount of money they spent in January.
Now, let's analyze the information given about their spending in February:
- Alison increased her spending by $140, so her spending in February is A + 140.
- Bryan doubled his spending, so his spending in February is 2B.
- Carly reduced her spending by $150, so her spending in February is C - 150.
According to the problem, they all spent the same amount of money in February. Therefore, we can set up the equation A + 140 = 2B = C - 150.
Now, let's find the value of each variable by solving the system of equations:
A + B + C = 1510
A + 140 = 2B
A + 140 = C - 150
First, let's simplify the second equation: A + 140 = 2B.
We can subtract 140 from both sides of the equation to isolate A: A = 2B - 140.
Now, substitute this expression for A in the first and third equations:
2B - 140 + B + C = 1510
2B - 140 + B = C - 150
Combine the like terms:
3B + C = 1650
3B = C - 10
Now, let's substitute the expression for C from equation 2 into equation 1:
3B + (2B - 140) = 1650
5B = 1790
B = 1790/5
B = 358
Substitute this value of B into equation 3 to find the value of C:
3(358) = C - 10
1074 = C - 10
C = 1074 + 10
C = 1084
Finally, substitute the values of B and C into equation 1 to find the value of A:
A + 358 + 1084 = 1510
A + 1442 = 1510
A = 1510 - 1442
A = 68
So, Alison spent $68, Bryan spent $358, and Carly spent $1084 in January.
Now, let's find the total amount of money Alison had in January at first. We are given that she saved $1850 in January. Savings is the difference between income and spending. Therefore, we can set up the equation:
Alison's total income - Alison's spending = Alison's savings
Alison's total income - 68 = 1850
Now, solve for Alison's total income:
Alison's total income = 1850 + 68
Alison's total income = 1918
Therefore, Alison had a total of $1918 in January at first.