Gopi loves cheeseburgers and hamburgers, and they have fat and calories as follows: cheeseburgers contain 13 13 grams of fat and 330 330 calories; hamburgers contain 18 18 grams of fat and 265 265 calories. How many cheeseburgers and hamburgers did Gopi eat when 111 111 grams of fat and 2050 2050 calories were consumed
Let's assume Gopi ate x cheeseburgers and y hamburgers.
From the given information, we can create two equations:
Equation 1: 13x + 18y = 111 (equation representing the total grams of fat consumed)
Equation 2: 330x + 265y = 2050 (equation representing the total calories consumed)
We can solve these equations simultaneously to find the values of x and y.
Multiply Equation 1 by 10 and Equation 2 by 3 to eliminate the decimals:
130x + 180y = 1110
990x + 795y = 6150
Multiply Equation 1 by 795 and Equation 2 by 180 to eliminate the y variable:
10335x + 14310y = 87930
59400x + 47700y = 369000
Now, subtract the second equation from the first equation:
(-59400x + 10335x) + (-47700y + 14310y) = 87930 - 369000
(-49065x) + (-33390y) = -281070
Divide the entire equation by -1 to simplify:
49065x + 33390y = 281070
Now, add this equation to Equation 2:
59400x + 47700y + 49065x + 33390y = 369000 + 281070
108465x + 81090y = 650070
Solve this equation for x and y using a calculator or algebraic methods:
x = 4
y = 3
Therefore, Gopi ate 4 cheeseburgers and 3 hamburgers.
Let's represent the number of cheeseburgers Gopi ate as "x" and the number of hamburgers as "y".
From the given information, we know that:
- Each cheeseburger contains 13 grams of fat and 330 calories.
- Each hamburger contains 18 grams of fat and 265 calories.
- The total fat consumed is 111 grams.
- The total calories consumed is 2050.
We can set up the following equations based on the given information:
Equation 1: 13x + 18y = 111 (Equation for total fat consumed)
Equation 2: 330x + 265y = 2050 (Equation for total calories consumed)
We can solve these equations to find the values of "x" and "y".
Multiplying Equation 1 by 265 and Equation 2 by 18 to eliminate the y terms:
265(13x) + 265(18y) = 265(111)
18(330x) + 18(265y) = 18(2050)
3455x + 4770y = 29415
5940x + 4770y = 36900
Now, subtracting the second equation from the first equation to eliminate y terms:
(3455x + 4770y) - (5940x + 4770y) = 29415 - 36900
-2485x = -7485
Dividing by -2485:
x = -7485 / -2485
x = 3
Now, substituting the value of x into Equation 1 to solve for y:
13x + 18y = 111
13(3) + 18y = 111
39 + 18y = 111
18y = 111 - 39
18y = 72
Dividing by 18:
y = 72 / 18
y = 4
Therefore, Gopi ate 3 cheeseburgers and 4 hamburgers when consuming 111 grams of fat and 2050 calories.
To solve this problem, we need to set up a system of equations based on the given information. Let's use c to represent the number of cheeseburgers and h to represent the number of hamburgers Gopi ate.
First, let's set up an equation for the total grams of fat consumed:
13c + 18h = 111
This equation represents the total grams of fat from the cheeseburgers (13c) and the total grams of fat from the hamburgers (18h) which should equal 111 grams.
Next, let's set up an equation for the total calories consumed:
330c + 265h = 2050
This equation represents the total calories from the cheeseburgers (330c) and the total calories from the hamburgers (265h) which should equal 2050 calories.
We now have a system of two equations with two unknowns:
13c + 18h = 111
330c + 265h = 2050
To solve this system, we can use the method of substitution or elimination. I will use the method of substitution to solve this system of equations.
We can start by solving the first equation for c:
13c = 111 - 18h
c = (111 - 18h) / 13
Now substitute this expression for c into the second equation:
330[(111 - 18h) / 13] + 265h = 2050
Simplify this equation to solve for h:
(330 * (111 - 18h)) / 13 + 265h = 2050
Multiply everything by 13 to get rid of the denominator:
330(111 - 18h) + 265h * 13 = 2050 * 13
Now expand and simplify the equation:
36630 - 5940h + 3445h = 26650
Combine like terms:
-2495h + 36630 = 26650
Rearrange the equation:
-2495h = 26650 - 36630
Simplify:
-2495h = -997980
Divide both sides by -2495:
h = -997980 / -2495
The result is:
h ≈ 400
So, it seems that Gopi ate approximately 400 hamburgers.
Now we can substitute this value of h back into the first equation to find the value of c:
13c + 18(400) = 111
13c + 7200 = 111
13c = -7089
Divide both sides by 13:
c = -7089 / 13
The result is:
c ≈ -545
It seems strange to have a negative number of cheeseburgers, so we need to consider that this may not be a feasible solution. It's possible that there is a mistake in the given information or in our calculations.
Please double-check your numbers and calculations to ensure accuracy.