How many pounds of pears did he buy?

Question

Mr. Adams buys a bag of apples and a bag of pears for a total of $14.00. There are eight pounds of fruit in all. The apples cost $1.50 per pound, and the pears cost $2.30 per pound

Answer 1

Okay to solve a question like this you need to use either the method of substitution or elimination. First we need to use the information we had to form two different equations. I will be using elimination in this instance but then after I'll show you how to use substitution for the same question.

The Elimination Method
Step 1: Define your variables.

Let x represent the number of pounds of apples.
Let y represent the number of pounds of pears.

Step 2: Make your equations.
1. x + y = 8 (the number of pounds of apples plus the number of pounds of pears equals 8 lbs)

2. 1.50x + 2.30y = 14 ($1.50 times the number of pounds of appls plus the $2.30 times the number of pounds of pears equals a total of $14)

Step 3: Eliminate a variable.

x + y = 8 ----------> 1.5x + 1.5y = 12
1.5x + 2.3y = 14 -----------> 1.5x + 2.30y = 14
It was easier to eliminate the x variable than the y, so what I did was I multiplied the first equation by 1.5 so that it would equal the 1.5 in the other equation and I can simply subtract out the x variable observe below:

1.5x + 1.5y = 12 SUBTRACT
1.5x + 2.30y = 14
-------------------------
0 + -0.8y = -2 Now we divide to isolate the y.
-0.8y/-0.8 = -2/-0.8
y = 2.5 <--- this is the answer for number of pounds of pears.

note: since 1.5 - 1.5 is 0 make sure u draw a line crossing out the x's and you dont have to write 0 + -0.8 I only did that to show you that x is gone.

Now are we done? No. We want to double check and make sure we got the answer so now let's solve for the number of pounds of apples.

Next step: Substitute y = 2.5 into the first equation.

x + y = 8
x+ (2.5) = 8
x = 8 - 2.5
x = 5.5 <---- number of pounds of apples

note: we moved the 2.5 over to the opposite side making it negative so we subtract it from the 8. Why did we move it over? To isolate the variable.

Now let's double check and make sure our answer satisfies the question.

x + y = 8
(5.5) + (2.5) = 8
8 = 8
LS = RS -our answer satisfies the first equation

1.50x + 2.30y = 14
1.5(5.5) + 2.3(2.5) = 14
8.25 + 5.75 = 14
14 = 14
LS = RS - r answer satisfies both equations thus it must be correct

Now to finish off your question by writing down your "therefore" statement which in this case is:

.: He bought 2.5lbs of pears

And the question is complete. But I will nowshow you how to do it with the method of substitution so you know for the future.

The Substition Method
Step 1 & 2 are the same as in elimination.

Step 3: Isolate a variable in one of the equations.
I am going to isolate the y in the first equation so we have:

x + y = 8
y = -x + 8 (x becomes negative b/e its been moved to the other side of the equal sign)

Step 4: Now substitute our isolated variable into the second equation (because we isolated the variable in the first, had we isolated something in the second equation we'd substitute it into the first)

So,

1.50x + 2.30y = 14
1.50x + 2.30(-x +8) = 14

Now work through the multiplication

1.50x - 2.30x +18.4 = 14
-0.8x = 14 -18.4 (we substracted our two x values and got -0.8x then we moved the 18.4 over to the other side of the equal sign to isolate the x making it change to a negative in the processs)

-0.8x = -4.4
-0.8x/-0.8 = -4.4/-0.8
x = 5.5 <---- number of pounds of apples

Now we must substitute x = 5.5 we will now work with the first equation again.

x + y = 8
(5.5) + y = 8
y = 8 - 5.5 (just so you know you could also do -5.5 + 8 and get the same answer as doing 8 - 5.5)
y = 2.5

.: He bought 2.5 lbs of pears

- the answer we got with substitution is the same as what we got with the elimination equation so we know its correct but if you were to use this method alone make sure you check your answer just like we did with the elimination method.

Substition method steps summary: isolate x or y in either equation, substitute that into the opposite equation than solve for the missing variable.

Answer 2

The first responder did a masterful job in explaining two methods. There is a third, known as marginal analysis.
If he bought only 8 lbs of apples, he would have paid $ 12. So for every pound of pears he buys in place of apples, he spends 80 cents more. Since he needs to buy $2 more, he has to buy 2.5 pounds of pears (2.00/0.80).