# Baseball Math Problem Solved the Hard Way (Algebra)

Question:

If the price of a bat and ball together is \$1.10 and the bat costs \$1 more than the ball.
Then how much is the ball?

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This problem is tricky. At a glance, one would just assume the ball will cost 10 cents since \$1.10 – \$1.00 is 0.10. But this is incorrect.

I’m sure this can be solved using intuition and simple arithmetic.
But here in Apptato we like doing things the hard way.

So let’s solve this using some algebra.

Let x be the price of the bat and y be the price of the ball.
We know that the price of the bat and ball together is \$1.10,
So we create the first equation: x + y = 1.10

And we also know that the price of the bat is \$1.00 more than the ball.
We can represent it in the second equation:
x – y = 1.00, which just means the price difference is \$1.00 with x (price of the bat) being the greater of the 2.

We now see a pair of equations known as simultaneous equations.

x + y = 1.10
x – y = 1.00

When there are 2 or more equations we are able to the sum of all the components of the left side and equate it to the sum of all the components of the right side. The equality is preserved.
The sum or summation is signified by the greek letter capital Sigma.

On the left side: x + x is 2x, y – y is 0
while on the right side: 1.10 + 1.00 = 2.10

2x = 2.10
x = 1.05

So the bat costs 1.05.

Substituting x back into either equation (let’s go with the second one)

x – y = 1.00
1.05 – y = 1.00
y = 0.05

This is just the same as saying: since the bat is 1.05 and is 1.00 more than the ball therefore the ball is \$0.05.

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