Mimi's Market sold 24 heads of lettuce one morning. That afternoon 2/7 of the remaining heads of lettuce were sold. The number of heads left was now 1/2 of the number the market had at the beginning of the day. How many heads of lettuce were there at the beginning of the day?
Let's solve this step-by-step:
Step 1: Calculate the number of heads of lettuce sold in the afternoon.
In the afternoon, 2/7 of the remaining heads of lettuce were sold.
So, the number of heads sold = (2/7) * (heads remaining) = (2/7) * (24 - heads sold in the morning).
Step 2: Set up an equation to represent the given information.
According to the problem, the number of heads remaining in the afternoon is 1/2 of the number of heads at the beginning of the day.
So, (24 - heads sold in the morning) - (2/7) * (24 - heads sold in the morning) = (1/2) * 24.
Step 3: Solve the equation from step 2.
(5/7) * (24 - heads sold in the morning) = (1/2) * 24.
To simplify,
5/7 * (24 - heads sold in the morning) = 12.
Multiply both sides by 7/5,
24 - heads sold in the morning = (7/5) * 12 = 84/5.
Subtract 24 from both sides,
heads sold in the morning = 84/5 - 24 = 84/5 - 120/5 = -36/5.
Step 4: Interpret the result.
Since the number of heads sold in the morning cannot be negative, this means there was a mistake in our calculations or in the given information. Please check the problem statement again or provide more information.
To solve this problem, we need to work backwards. Let's represent the unknown number of lettuce heads Mimi's Market had at the beginning of the day as "x".
According to the problem, Mimi's Market sold 24 heads of lettuce in the morning. So, the remaining lettuce heads after the morning sale would be "x - 24".
In the afternoon, 2/7 of the remaining heads of lettuce were sold. This means 2/7 * (x - 24) were sold and the remaining heads would be (x - 24) - 2/7 * (x - 24).
According to the problem, the number of heads left after the afternoon sale was 1/2 of the number Mimi's Market had at the beginning of the day. So, we can set up an equation:
(x - 24) - 2/7 * (x - 24) = 1/2 * x
Now, let's solve for x.
Multiply both sides of the equation by 2 to eliminate the fraction:
2 * [(x - 24) - 2/7 * (x - 24)] = 2 * (1/2 * x)
Simplifying the equation:
2 * (x - 24 - 2/7 * (x - 24)) = x
Distribute 2 to both terms inside the parentheses:
2 * x - 48 - 4/7 * (x - 24) = x
Simplify further:
2x - 48 - 4/7x + 8/7 * 24 = x
Combine like terms:
2x - 48 - 4/7x + 192/7 = x
Multiply all terms by 7 to eliminate the fraction:
14x - 336 - 4x + 192 = 7x
Combine like terms:
10x - 144 = 7x
Move 7x to one side by subtracting it from both sides:
10x - 7x - 144 = 0
Simplify:
3x - 144 = 0
Add 144 to both sides:
3x = 144
Divide both sides by 3:
x = 48
Therefore, there were 48 heads of lettuce at the beginning of the day.
X = # heads to begin the day.
(X-24) = Bal. after morning sale.
(2/7)*(X-24) sold during afternoon.
Bal. = (5/7)*(X-24) = X/2
(10/7)(X-24) = X
(10x-240)/7 = X
10x - 240 = 7x
10x - 7x = 240
3x = 240
X = 80