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Question 476566: Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems. Some problems are worth 2 points and some problems are worth 4 points. How many problems of each value are on the test?)
Found 7 solutions by Theo, ikleyn, josgarithmetic, greenestamps, mccravyedwin, Edwin McCravy, AnlytcPhil:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x = number of questions that are 2 points each.
let y = number of questions that are 3 points each.
since the total number of questions is 35, then your first equation is:
x + y = 35
since each x gets you 2 points and each y gets you 3 points, then your ssecond equation is:
2x + 3y = 100
solve these 2 equations simultaneously for your answer.
use second equation to solve for y in terms of x.
you get y = 35 - x
substitute for y in the first equation to get:
2x + 3(35-x) = 100 which becomes:
2x + 105 - 3x = 100 which becomes:
-x + 105 = 100
add x to both sides of this equation and subtract 100 from both sides of this equation to get:
x = 5
since x + y = 35, this means that y = 30
you have 5 questions that are 2 points each and 30 questions that are 3 points each for a total of 5 + 30 = 35 questions with a total of 10 + 90 = 100 points.
Answer by ikleyn(52914)  (Show Source):
You can put this solution on YOUR website! .
Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems.
Some problems are worth 2 points and some problems are worth 4 points. How many problems of each value are on the test?)
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The solution in the post by @Theo is incorrect.
His equations are not adequate to the problem, and the answer is wrong.
I came to bring a correct solution.
Let x be the number of the 2-points problems and
let y be the number of the 4-points problems.
Write equations as you read the problem
x + y = 35, (1)
2x + 4y = 100. (2)
It can be solved by different methods: by the Substitution method or by the Elimination method.
I will solve it here using the Elimination method.
Multiply equation (1) by 2 (both sides). Keep equation (2) as is.
You will get
2x + 2y = 70, (1')
2x + 4y = 100. (2')
Now subtract equation (') from equation (2').
The terms with '2x' will annihilate each other, and you will get
4y - 2y = 100 - 70,
2y = 30,
y = 30/2 = 15.
So, y = 15. To find x, substitute this value y = 15 in equation (1).
You will get
x + 15 = 35 ---> x = 35 - 15 = 20.
ANSWER. There are 20 2-points problems and 15 4-points problems.
CHECK. Checking is a necessary part of the solution.
Substitute the found values x and y into equation (1) to make sure
that the left side value coincides with the right side value.
Substitute the found values x and y into equation (2) to make sure
that the left side value coincides with the right side value.
At this point, the solution is completed in full.
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There is no any need to recheck the @josgarithmetic' solution,
since he simply re-wrote it from my solution, with variations.
Answer by josgarithmetic(39630)  (Show Source):
Answer by greenestamps(13215)  (Show Source):
You can put this solution on YOUR website!
Here is a way to solve the problem mentally, using logical reasoning instead of formal algebra. The calculations performed are nearly identical to those used in the formal algebraic solution shown by the other tutor.
(1) If all 35 questions were worth 2 points each, the total number of points would be 35*2 = 70, which is 30 less than the actual total of 100. So we need another 30 points.
(2) Each 4-point question is worth 2 points more than each 2-point question.
(3) To get the additional 30 points, the number of 4-point questions needed is 30/2 = 15.
(4) So there are 15 4-point questions and 35-15 = 20 2-point questions.
ANSWER: 15 4-point questions and 20 2-point questions.
CHECK: 15(4) + 20(2) = 60+40 = 100
Answer by mccravyedwin(409)  (Show Source):
You can put this solution on YOUR website!
I thought you might enjoy comparing Greenestamps' solution to this problem:
Mrs. Jones has 35 animals, all sheep and ducks, on her farm. The animals
have a total of 100 legs and 35 heads. How many sheep and how many ducks
are on her farm?
(1) If all 35 animals were ducks, the total number of legs would be only
35*2 = 70, which is 30 legs less than the actual total of 100 legs.
So we need another 30 legs.
(2) Each sheep has 2 more legs than each duck.
(3) To get the additional 30 legs, the number of sheep needed is 30/2 = 15.
(4) So there are 15 sheep and 35-15 = 20 ducks.
ANSWER: 15 sheep and 20 ducks.
CHECK: 15(4) + 20(2) = 60+40 = 100
Edwin
Answer by Edwin McCravy(20065)  (Show Source):
You can put this solution on YOUR website!
You might also enjoy comparing this other solution to that problem:
Mrs. Jones has 35 animals, all sheep and ducks, on her farm. The animals
have a total of 100 legs and 35 heads. How many sheep and how many ducks
are on her farm?
(1) If all 35 animals were sheep, the total number of legs would be
35*4 = 140, which is 40 legs more than the actual total of 100 legs.
So we need to reduce the number of legs by 40.
(2) Each duck has 2 legs .
(3) To reduce the number of legs by an additional 40 legs, the number of
ducks needed is 40/2 = 20 ducks.
(4) So there are 20 ducks and 35-20 = 15 sheep.
ANSWER: 15 sheep and 20 ducks.
CHECK: 15(4) + 20(2) = 60+40 = 100
Edwin
Answer by AnlytcPhil(1810)  (Show Source):
You can put this solution on YOUR website!
josgarithmetic(39630) misread the 4 as a 3
Here is a correction of his solution, for his benefit.
let x = number of questions that are 2 points each.
let y = number of questions that are 4 points each.
since the total number of questions is 35, then your first equation is:
x + y = 35
since each x gets you 2 points and each y gets you 4 points, then your second equation is:
2x + 4y = 100
solve these 2 equations simultaneously for your answer.
use second equation to solve for y in terms of x.
you get y = 35 - x
substitute for y in the first equation to get:
2x + 4(35-x) = 100 which becomes:
2x + 140 - 4x = 100 which becomes:
-2x + 140 = 100
add 2x to both sides of this equation....
140 = 100 +2x
and subtract 100 from both sides of this equation to get:
40 = 2x
20 = x
since x + y = 35, this means that y = 15
you have 20 questions that are 2 points each and 15 questions that are 4 points each for a total of 20 + 15 = 35 questions with a total of 40 + 60 = 100 points.
Edwin
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