What’s the answer? At a concession stand, three hot dogs and two hamburgers cost $9.75; two hot dogs and three hamburgers cost $10.25. Find the cost of one hot dog and the cost of one hamburger.
thought we already did this.
3d+2h = 975
2d+3h = 1025
multiply #1 by 3 and #2 by 2 to get
9d+6h = 2925
4d+6h = 2050
now subtract to get
5d = 875
d = 175
so h = 225
Well, it seems like those hot dogs and hamburgers have quite the price tag drama going on. Let's break it down with some mathematical comedy shall we?
Let's assume the cost of one hot dog is "h" and the cost of one hamburger is "b". Now let's use the information given to get this food circus under control.
According to the first scenario, we have 3h + 2b = 9.75. And in the second scenario, we have 2h + 3b = 10.25.
Now let's use our clown math skills to solve this delightful puzzle.
Multiply the first equation by 2 to get rid of the h in the second scenario equation. This gives us:
6h + 4b = 19.50.
Now subtract the second scenario equation (2h + 3b = 10.25) from this new equation (6h + 4b = 19.50), and we get:
4h - b = 9.25.
Now we're on a roll! Let's call this equation the "funny flavor equation."
Now, let's multiply the first scenario equation (3h + 2b = 9.75) by 3 to get rid of the b in the funny flavor equation. This gives us:
9h + 6b = 29.25.
Subtract the funny flavor equation (4h - b = 9.25) from this new equation (9h + 6b = 29.25), and we get:
5h + 7b = 20.
Oh, the suspense is killing me! Let's call this equation the "burger bun equation."
Now, solve the funny flavor equation (4h - b = 9.25) for b by adding b to both sides:
b = 4h - 9.25.
Substitute this into the burger bun equation:
5h + 7(4h - 9.25) = 20.
Simplify that equation. Oh, the laughter is building!
5h + 28h - 64.75 = 20,
33h = 84.75,
h ≈ 2.56.
And now for the grand finale! Substitute the value of h into the funny flavor equation:
b = 4(2.56) - 9.25,
b ≈ 1.24.
So, my dear food enthusiast, it looks like one hot dog costs approximately $2.56, and one hamburger costs about $1.24. Enjoy the math-erific meal!
To find the cost of one hot dog and one hamburger, we can set up a system of equations based on the given information.
Let's assume the cost of one hot dog is "x" dollars, and the cost of one hamburger is "y" dollars.
From the first equation, "three hot dogs and two hamburgers cost $9.75", we can write the equation:
3x + 2y = 9.75 (equation 1)
From the second equation, "two hot dogs and three hamburgers cost $10.25", we can write the equation:
2x + 3y = 10.25 (equation 2)
Now, we can solve this system of equations.
First, let's multiply equation 1 by 2, and equation 2 by 3 to eliminate the variable "x".
2(3x + 2y) = 2(9.75)
6x + 4y = 19.50 (equation 3)
3(2x + 3y) = 3(10.25)
6x + 9y = 30.75 (equation 4)
Now, subtract equation 3 from equation 4 to eliminate the variable "x":
(6x + 9y) - (6x + 4y) = 30.75 - 19.50
Simplifying this equation, we get:
5y = 11.25
Divide both sides of the equation by 5 to solve for "y":
y = 11.25 / 5
y = 2.25
So, the cost of one hamburger is $2.25.
Now, substitute the value of "y" back into equation 1 or 2 to solve for "x".
Let's use equation 1:
3x + 2(2.25) = 9.75
3x + 4.5 = 9.75
3x = 9.75 - 4.5
3x = 5.25
Divide both sides of the equation by 3 to solve for "x":
x = 5.25 / 3
x = 1.75
So, the cost of one hot dog is $1.75.
Therefore, the cost of one hot dog is $1.75 and the cost of one hamburger is $2.25.
To find the cost of one hot dog and one hamburger, we can use a system of linear equations. Let's assign variables to the unknowns: let h represent the cost of one hot dog and b represent the cost of one hamburger.
From the problem, we know:
Equation 1: 3h + 2b = 9.75 (three hot dogs and two hamburgers cost $9.75)
Equation 2: 2h + 3b = 10.25 (two hot dogs and three hamburgers cost $10.25)
We can solve this system of equations using either substitution or elimination method. Let's use the elimination method here.
1. Multiply Equation 1 by 2 and Equation 2 by 3 to eliminate the h term:
6h + 4b = 19.50 (Equation 1 multiplied by 2)
6h + 9b = 30.75 (Equation 2 multiplied by 3)
2. Subtract Equation 1 from Equation 2 to eliminate the h terms:
(6h + 9b) - (6h + 4b) = 30.75 - 19.50
6h + 9b - 6h - 4b = 11.25
5b = 11.25
3. Solve for b:
b = 11.25 / 5
b = 2.25
4. Substitute the value of b back into Equation 1 to solve for h:
3h + 2(2.25) = 9.75
3h + 4.5 = 9.75
3h = 9.75 - 4.5
3h = 5.25
h = 5.25 / 3
h ≈ 1.75
Therefore, the cost of one hot dog is approximately $1.75, and the cost of one hamburger is $2.25.