Math  /  Algebra

QuestionTen mangos and twelve avocadoes cost $23\$ 23. Five mangos and four avocadoes cost $10\$ 10. How much do one mango and one avocado cost? \$ 1.4

Studdy Solution

STEP 1

What is this asking? We need to find the price of one mango and one avocado, given the total cost of two different combinations of mangos and avocados. Watch out! Don't mix up the number of fruits and their prices!
We need to be organized to keep track of everything.

STEP 2

1. Set up the equations
2. Solve for one variable
3. Solve for the other variable
4. Calculate the combined cost

STEP 3

Alright, let's translate the problem into math!
Let mm be the cost of one mango and aa be the cost of one avocado.
We're dealing with two situations here.

STEP 4

First, ten mangos and twelve avocados cost $23\$23.
We can write this as an equation: 10m+12a=2310m + 12a = 23.

STEP 5

Second, five mangos and four avocados cost $10\$10.
This gives us another equation: 5m+4a=105m + 4a = 10.

STEP 6

Now, we have two equations with two unknowns.
Let's solve for one of the variables.
We can **multiply** the second equation by 2-2.
Why 2-2?
Because this will give us 10m-10m, which will **add to zero** with the 10m10m in the first equation!

STEP 7

Multiplying the second equation by 2-2 gives us 10m8a=20-10m - 8a = -20.

STEP 8

Now, let's **add** this new equation to the first equation: 10m+12a=2310m8a=204a=3\begin{array}{rcl} 10m + 12a & = & 23 \\ -10m - 8a & = & -20 \\ 4a & = & 3 \end{array}

STEP 9

So, 4a=34a = 3.
Now, we can **divide** both sides by 44 to **divide to one**, giving us a=34a = \frac{3}{4}, which is $0.75\$0.75.
So, one avocado costs $0.75\$0.75!

STEP 10

Great! Now that we know the cost of one avocado, let's substitute a=0.75a = 0.75 back into one of our original equations.
Let's use the second equation, 5m+4a=105m + 4a = 10, because it looks a little simpler.

STEP 11

Substituting a=0.75a = 0.75 into the equation gives us 5m+40.75=105m + 4 \cdot 0.75 = 10.

STEP 12

Simplifying, we get 5m+3=105m + 3 = 10.

STEP 13

Subtracting 33 from both sides gives us 5m=75m = 7.

STEP 14

Finally, **divide** both sides by 55 to **divide to one**, giving us m=75m = \frac{7}{5}, which is $1.40\$1.40.
One mango costs $1.40\$1.40!

STEP 15

The problem asks for the combined cost of one mango and one avocado.
We know that one mango costs $1.40\$1.40 and one avocado costs $0.75\$0.75.

STEP 16

Adding these together, we get $1.40+$0.75=$2.15\$1.40 + \$0.75 = \$2.15.

STEP 17

One mango and one avocado cost $2.15\$2.15.

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