Over the last three evenings, Martina received a total of 83 calls at the call center. The second evening, she received 4 times as many calls as the third evening. The first evening, she received 7 fewer calls than the third evening. How many phone calls did she receive each evening?
Let's call the number of calls Martina received on the first evening "x".
We know that she received 4 times as many calls on the second evening as she did on the third, so if we call the number of calls she received on the third evening "y", then Martina received 4y calls on the second evening.
We also know that over three evenings, she received a total of 83 calls, so:
x + 4y + y = 83
Simplifying this equation, we get:
x + 5y = 83
We also know that Martina received 7 fewer calls on the first evening than she did on the third:
x = y - 7
We can substitute this expression for "x" in the equation we derived earlier:
(y - 7) + 5y = 83
Simplifying this equation, we get:
6y - 7 = 83
Adding 7 to both sides, we get:
6y = 90
Dividing both sides by 6, we get:
y = 15
So Martina received 15 calls on the third evening. Using the expression we derived earlier for "x", we can find that she received 8 calls on the first evening (since x = y - 7, and y = 15).
Finally, we can use the fact that Martina received 4 times as many calls on the second evening as she did on the third to find that she received 60 calls on the second evening (since 4y = 4 * 15 = 60).
So overall, Martina received 8 calls on the first evening, 60 calls on the second evening, and 15 calls on the third evening.
Let's break down the information given:
1. The total number of calls over the last three evenings is 83.
2. The second evening, Martina received 4 times as many calls as the third evening.
3. The first evening, she received 7 fewer calls than the third evening.
Let's assign variables for each evening:
Let x = the number of calls Martina received on the first evening.
Let y = the number of calls Martina received on the second evening.
Let z = the number of calls Martina received on the third evening.
We can set up equations based on the given information:
1. x + y + z = 83 (the total number of calls over the three evenings is 83).
2. y = 4z (the second evening, she received 4 times as many calls as the third evening).
3. x = z - 7 (the first evening, she received 7 fewer calls than the third evening).
Now, let's solve these equations step by step:
Step 1: Substitute equations (2) and (3) into equation (1) to eliminate y and z:
z - 7 + 4z + z = 83
6z - 7 = 83
Step 2: Add 7 to both sides of the equation to isolate the variable:
6z = 90
Step 3: Divide both sides of the equation by 6 to solve for z:
z = 15
Step 4: Substitute the value of z into equation (2) to find y:
y = 4 * 15
y = 60
Step 5: Substitute the value of z into equation (3) to find x:
x = 15 - 7
x = 8
Therefore, Martina received 8 calls on the first evening, 60 calls on the second evening, and 15 calls on the third evening.