three apples and 2 ponkans cost P51. two apples and 3 ponkans cost P46.50. how much will half a dozen apple and half a dozen ponkans cost?

one way:

3a+2p = 51
2a+3p = 46.50
add them up:
5a+5p = 97.50
multiply by 6/5:
6a+6p = 97.0 * 6/5 = 117.00
note that we didn't have to know how much each item cost.

another way:

multiply 1st by 2 and 2nd by 3 to get
6a+4p = 102
6a+9p = 139.50
subtract 1st from 2nd to get
5p = 37.50
p = 7.50
a = 12.00
6a+6p = 72.00 + 45.00 = 117.00

thanks so much... but what if the cost of each ponkan and apples is asked? what will be the solution?

excuse me???

look at the second solution, where I actually figure the cost of each.

yeah its there...thanks a lot...

To solve this problem, let's assign variables to the cost of one apple and one ponkan. Let's say the cost of one apple is 'x' and the cost of one ponkan is 'y'.

According to the first statement, three apples and 2 ponkans cost P51. So we can write the equation as:
3x + 2y = 51.

According to the second statement, two apples and 3 ponkans cost P46.50. So we can write the equation as:
2x + 3y = 46.50.

Now, we have a system of equations that we can solve to find the values of 'x' and 'y'. Let's solve them using the substitution method:

From the first equation, we can solve for x:
3x = 51 - 2y
x = (51 - 2y) / 3.

Substituting this value of x into the second equation:
2((51 - 2y) / 3) + 3y = 46.50,
(102 - 4y) / 3 + 3y = 46.50,
102 - 4y + 9y = 139.50,
5y = 37.50,
y = 37.50 / 5,
y = 7.50.

Now, substituting the value of y back into the first equation:
3x + 2(7.50) = 51,
3x + 15 = 51,
3x = 51 - 15,
3x = 36,
x = 36 / 3,
x = 12.

So, the cost of one apple (x) is P12 and the cost of one ponkan (y) is P7.50.

Now, let's calculate the cost of half a dozen apples (6 apples) and half a dozen ponkans (6 ponkans):

Cost of half a dozen apples = 6 * 12 = P72.
Cost of half a dozen ponkans = 6 * 7.50 = P45.

Therefore, half a dozen apples and half a dozen ponkans will cost P72 + P45 = P117.