Thread: Solving Problems with Linear Equations

Holiday homework strikes again!

1) A wine merchant buys 50 cases of wine. He pays full price for half of them but gets a 40% discount on the remainder. If he paid a total of $2260, how much was the full price of a single case?

2) A rectangular courtyard is three times as long as it is wide. If the perimeter of the courtyard is 67 m, find the dimensions of the courtyard.

3) Three girls are playing Scrabble. At the end of the game, the total of their scores adds up to 504. Annie scored 10% more than Belinda, while Cassie scored 60% of the combined scores of the other two. What did each player score?

4) The mass of a molecule of a certain chemical compound is 2.45 x 10^-22 g. If each molecule is made up of two carbon atoms and six oxygen atoms and the mass of one oxygen atom is (1/3) that of a carbon atom, find the mass of an oxygen atom.

Any help would be much appreciated.

Originally Posted by Fratricide

Holiday homework strikes again!

1) A wine merchant buys 50 cases of wine. He pays full price for half of them but gets a 40% discount on the remainder. If he paid a total of $2260, how much was the full price of a single case?

Let P be the full price. Half of 50 is 25 so he paid 25P for them. A 40% discount leaves a price of 60%, or .60P for the other 25 so he paid (25)(.6P) for them. Add those and set the sum equal to 2260.

2) A rectangular courtyard is three times as long as it is wide. If the perimeter of the courtyard is 67 m, find the dimensions of the courtyard.

The perimeter of a rectangle is 2 times the length plus two times the length. If you call the length W, the length is 3W. What equation do you get from that?

3) Three girls are playing Scrabble. At the end of the game, the total of their scores adds up to 504. Annie scored 10% more than Belinda, while Cassie scored 60% of the combined scores of the other two. What did each player score?

Let "B" be Belinda's score. Then Annie's score if (1+ 0.1)B= 1.1B (do you see why?). That means that the "combined score of the other two" is B+ 1.1B= 2.1B and Cassies score is 60% of that- .6(2.1B)=12.6B. Add those and set them equal to 504.

4) The mass of a molecule of a certain chemical compound is 2.45 x 10^-22 g. If each molecule is made up of two carbon atoms and six oxygen atoms and the mass of one oxygen atom is (1/3) that of a carbon atom, find the mass of an oxygen atom.

Since you are asked for the mass of an oxygen atom, let "M" represent that mass. Then the mass of a a carbon atom is 3M. The mass of whole molecule is 2(3M)+ 6M= 6M+ 6M= 12M. Set that equal to 2.45 x 10^-22 and solve for M.

Any help would be much appreciated.

1) Let price of each be x
then cost of 25 = 25x,
cost of remaining 25 = 25x × 0.6 = 15x
25x + 15x = 5560
40x = 5560
x = $ 139 for each case.
2) length = 3 × width
L = 3w
perimeter = 2(L + w) = 67
2 × 4w = 67
w = 67/8 = 8.375 m
L = 3 × 8.375 = 25.125 m
3) Three girls are playing Scrabble. At the end of the game, the total of their scores adds up to 504. Annie scored 10% more than Belinda, while Cassie scored 60% of the combined scores of the other two. What did each player score?
Annie scored a Belinda scored b cassie scored c
a + b + c = 504
a = 1.1b
c = 0.6(a + b) = 0.6a + 0.6b

a + b + c = 504
1.1b + b + 0.6(1.1b) + 0.6b = 504
1.1b + b + 0.66b + 0.6b = 504
3.36b = 504
b = 150
a = 165
c = 189

4) 2c + 6o = 2.45 × 10^-22
2c + 6(c/3) = 2.45 × 10^-22
4c = 2.45 × 10^-22
c = 0.6125 × 10^-22
o = 0.6125/3 × 10-22
= 0.2042 × 10^-22
= 2.042 × 10^-26 g
----

Thanks heaps!

Learning from those problems I managed to complete most of the others, save for these two:

1) Bronwyn and Noel have a women's clothing shop in Summerland and Bronwyn manages the shop. Sally, Adam and Baby Lana came into the shop and Sally bought dresses and handbags. The dresses cost $65 each and the handbags cost $36 each. The total number of items was 11 and in total she spent $598. How many dresses and how many handbags did she buy?

I had something along the lines of "65d + 36h = 598" but I don't know where to go from there.

2) A real restate agent sells 22 houses in six months. He makes a commission of $11,500 per house on some and $13,000 on the remainder. If his total commission over the six months was $272,500, on how many houses did he make a commission of $11,500?

Originally Posted by Fratricide

Thanks heaps!

Learning from those problems I managed to complete most of the others, save for these two:

1) Bronwyn and Noel have a women's clothing shop in Summerland and Bronwyn manages the shop. Sally, Adam and Baby Lana came into the shop and Sally bought dresses and handbags. The dresses cost $65 each and the handbags cost $36 each. The total number of items was 11 and in total she spent $598. How many dresses and how many handbags did she buy?

I had something along the lines of "65d + 36h = 598" but I don't know where to go from there.

It is impossible to say anything about that because you did not tell us what "d" and "h" are supposed to mean. If "d" is "number of dresses" and "h" is "number handbags" the, yes, the equation you give has the formula for calculating the total amount of money spend on the left and the actual money spent on the right so they are equal. Now if you know she bought "d" dresses and "h" handbags, how would you find how many total items she bought? You are told that is equal to 11. Solve those two equations.

2) A real restate agent sells 22 houses in six months. He makes a commission of $11,500 per house on some and $13,000 on the remainder. If his total commission over the six months was $272,500, on how many houses did he make a commission of $11,500?

Let x be the number of houses on which he earned a commission of $11500 (which is what you want to find) and y be the number of houses on which he earned a commission of $13000. What is x+ y equal to? What is 11500x+ 13000y equal to? Solve those two equations.

1) Bronwyn and Noel have a women's clothing shop in Summerland and Bronwyn manages the shop. Sally, Adam and Baby Lana came into the shop and Sally
bought dresses = d and handbags = h total cost 598 seems to be 599
cost = 65d + 36h = 599 ----------- (1)
The total number of items d + h = 11 ----------- (2) -- × 36
36d + 36h = 396 ---------- (3)
(1) – (3)
29d = 203
d = 7 and h = 4

2) Let types of house be a and b
a + b = 22
11500a + 13000b = 272500
11500a + 11500b = 253000
1500b = 19500
b = 13
a = 9
-----

1) Bronwyn and Noel have a women's clothing shop in Summerland and Bronwyn manages the shop. Sally, Adam and Baby Lana came into the shop and Sally
bought dresses = d and handbags = h total cost 598 seems to be 599
cost = 65d + 36h = 599 ----------- (1)
The total number of items d + h = 11 ----------- (2) -- × 36
36d + 36h = 396 ---------- (3)
(1) – (3)
29d = 203
d = 7 and h = 4

2) Let types of house be a and b
a + b = 22
11500a + 13000b = 272500
11500a + 11500b = 253000
1500b = 19500
b = 13
a = 9
-----

Thanks again.