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Grade 7/8 POTW We also did this one last year, but I'll do it again. First, let's list the facts. Note* I already made the sums and the variables that add up to them: - a<b<c - a + b = 2986 - a + c = 3464 - b + c = 3550 I'm just going to go straight into the math. (a+c) - (a+b) = c - b = 3464 - 2986 = 478 (b+c) - (b+a) = c - a = 3550 - 2986 = 564 (c+b) - (c+a) = b - a = 3550 - 3464 = 86 Therefore: c = b + 478 c = a + 564 b = a + 86 b = c - 478 a = c - 564 etc (many equations can be formed) We can now use substitution and algebra to solve. a: 2986 = a + (a + 86) 2986 = 2a + 86 2986 - 86 = 2a 2900 = 2a 1450 = a b: 3550 = b + (b + 478 3550 = 2b + 478 3550 - 478 = 2b 3072 = 2b 1536 = b c: 3464 = c + (c - 564) 3464 = 2c - 564 3464 + 564 = 2c 4028 = 2c 2014 = c a = 1450 b = 1536 c = 2014 To double-check with the actual values, we can simply just plug in the values into the equation. a + b = 1450 + 1536 = 2986 a + c = 1450 + 2014 = 3464 b + c = 1536 + 2014 = 3550 Therefore, the value of the largest value is 2014.
Grade 7/8 POTW: Info a<b<c When added, gives 2986, 3464 and 3550
Using common sense, I know that because of larger and smaller numbers, the following equations must be true. a+b=2986 a+c=3464 b+c=3550
The question is basically asking “what is the largest possible value for c. We need a to be big, but not too big because we need b to be small, so c can be large. a has to be a large number but small enough to be smaller than b. To find a possibility, we can divide the same 2986 by 2 and make a one smaller than the quotient. 2986/2=1493. 1493-1=1492. a=1492 b=1494 c= 3550-1494= 2056
This is the largest possible value for the largest number. (If I did anything wrong, I’ll fix any errors).
We can try to combine all three equations into one larger equation: (a+b)+(b+c)+(a+c)=2986+3464+3550 (Since this is addition, we can open the brackets) 2a+2b+2c=10000 (Factor out the common factor) 2(a+b+c)=10000 a+b+c=5000
To find c, we can take what we know about the sum of all three numbers and combine that with the equation of a and b. (a+b+c)-(a+b)=5000-2986 (Since we are now dealing with subtraction of a bracket, we need to flip the sign from inside the bracket). a+b+c+-a-b=2014 (Combine like terms) c=2014
a= 3464-2014 a=1450
b=3550-2014 b=1536
(therefore) the largest number is 2014.
Correct me if I’m wrong, but I also remember doing this POTW last year.
Grade 7/8 POTW If B and C are the two largest numbers, then their sum has to be equal to the largest number which is 3550. If A and C are the largest and smallest number, they have to be equal to the second largest number which is 3464. Lastly, A and B are the smallest and middle number meaning that their sum has to be equal to 2986. I will use guess and check to solve this problem and my first guess will correspond with the given amounts that the letters are measured to approximately be. After about 2 guesses I was left with the correct answer of c=2014 b=1536 a=1450 My first guess was c=2000 and then with this amount, I found out the values of the other letters and concluded that the answer was about 40 off. I then raised and raised the value of C until I came up with 2014 which was the correct value of C. Then, I found out the other values to get my answer.
The largest number is 2014 and I only found out the other values to check that my answer was correct.
We know that a+b subtracted from a+b+c must equal the value of c. To find this, we need to find out the value of a+b+c, and subtract the simultaneous equations. We know that (a+b) = 2986, (a+c) = 3464 and (b+c)= 3550. Adding these up, we get pais of both a, b, and c. Then, we just subtract.
2a+2b+2c=2986+3464+3550=10000 - 2a+2b=5972 = 2c=4028 c=2014 Then, we can do the operations to find the value of b and a. 2014+b=3550 b=3550-2014 b=1536 2014+a=3464 a=3464-2014 a=1450 DOUBLE CHECK a+b=2986 1450+1536=2986 is true
Therefore, the biggest number is 2014, the smallest number is 1450 and the middle number is 1536.
a+b=smallest=2986 c+a=middle number=3464 b+c= the largest=3550 2986(a+b)+3464(a+c)+3550(b+c)=1000 2a+2b+2c= 1000 1000/2= 500 1000/2= a+b+c 500=a+b+c 500-a+b=c 500-2986=c c=2014 The answer for c is 2014
We know that a ＜ b ＜ c, and: a + b = 2986 - Equation 1 a + c = 3464 - Equation 2 b + c = 3550 - Equation 3 This means that: b + (3464 - a) = 3550 - From equations 2 and 3 b - a = 3550 - 3464 = 86 b - a = 86 b = a + 86 Now, using Equation 1 a + (a + 86) = 2986 2a = 2986 - 86 Therefore, a = 2900 / 2 = 1450 Now, we know that a = 1450 Therefore, b = a + 86 = 1450 + 86 = 1536 We also know that b = 1536 Finally, 1536 + c = 3550, so 3550 - 1536 = 2014 We know that c = 2014 So, 1450 ＜ 1536 ＜ 2014
Grade 7/8 POTW
ReplyDeleteWe also did this one last year, but I'll do it again. First, let's list the facts. Note* I already made the sums and the variables that add up to them:
- a<b<c
- a + b = 2986
- a + c = 3464
- b + c = 3550
I'm just going to go straight into the math.
(a+c) - (a+b) = c - b = 3464 - 2986 = 478
(b+c) - (b+a) = c - a = 3550 - 2986 = 564
(c+b) - (c+a) = b - a = 3550 - 3464 = 86
Therefore:
c = b + 478
c = a + 564
b = a + 86
b = c - 478
a = c - 564
etc (many equations can be formed)
We can now use substitution and algebra to solve.
a:
2986 = a + (a + 86)
2986 = 2a + 86
2986 - 86 = 2a
2900 = 2a
1450 = a
b:
3550 = b + (b + 478
3550 = 2b + 478
3550 - 478 = 2b
3072 = 2b
1536 = b
c:
3464 = c + (c - 564)
3464 = 2c - 564
3464 + 564 = 2c
4028 = 2c
2014 = c
a = 1450
b = 1536
c = 2014
To double-check with the actual values, we can simply just plug in the values into the equation.
a + b = 1450 + 1536 = 2986
a + c = 1450 + 2014 = 3464
b + c = 1536 + 2014 = 3550
Therefore, the value of the largest value is 2014.
If you're in Grade 8 then you might see a trend here. Not every week of course. And yes, do them again if it is the case!
DeleteGrade 7/8 POTW:
ReplyDeleteInfo a<b<c
When added, gives 2986, 3464 and 3550
Using common sense, I know that because of larger and smaller numbers, the following equations must be true.
a+b=2986
a+c=3464
b+c=3550
The question is basically asking “what is the largest possible value for c. We need a to be big, but not too big because we need b to be small, so c can be large. a has to be a large number but small enough to be smaller than b. To find a possibility, we can divide the same 2986 by 2 and make a one smaller than the quotient. 2986/2=1493. 1493-1=1492.
a=1492
b=1494
c= 3550-1494= 2056
This is the largest possible value for the largest number. (If I did anything wrong, I’ll fix any errors).
We can try to combine all three equations into one larger equation:
ReplyDelete(a+b)+(b+c)+(a+c)=2986+3464+3550 (Since this is addition, we can open the brackets)
2a+2b+2c=10000 (Factor out the common factor)
2(a+b+c)=10000
a+b+c=5000
To find c, we can take what we know about the sum of all three numbers and combine that with the equation of a and b.
(a+b+c)-(a+b)=5000-2986 (Since we are now dealing with subtraction of a bracket, we need to flip the sign from inside the bracket).
a+b+c+-a-b=2014 (Combine like terms)
c=2014
a= 3464-2014
a=1450
b=3550-2014
b=1536
(therefore) the largest number is 2014.
Correct me if I’m wrong, but I also remember doing this POTW last year.
If you're in Grade 8 then you might see a trend here. Not every week of course.
DeleteGrade 7/8 POTW
ReplyDeleteIf B and C are the two largest numbers, then their sum has to be equal to the largest number which is 3550. If A and C are the largest and smallest number, they have to be equal to the second largest number which is 3464. Lastly, A and B are the smallest and middle number meaning that their sum has to be equal to 2986. I will use guess and check to solve this problem and my first guess will correspond with the given amounts that the letters are measured to approximately be.
After about 2 guesses I was left with the correct answer of
c=2014
b=1536
a=1450
My first guess was c=2000 and then with this amount, I found out the values of the other letters and concluded that the answer was about 40 off. I then raised and raised the value of C until I came up with 2014 which was the correct value of C. Then, I found out the other values to get my answer.
The largest number is 2014 and I only found out the other values to check that my answer was correct.
We know that a+b subtracted from a+b+c must equal the value of c. To find this, we need to find out the value of a+b+c, and subtract the simultaneous equations. We know that (a+b) = 2986, (a+c) = 3464 and (b+c)= 3550. Adding these up, we get pais of both a, b, and c. Then, we just subtract.
ReplyDelete2a+2b+2c=2986+3464+3550=10000
-
2a+2b=5972
=
2c=4028
c=2014
Then, we can do the operations to find the value of b and a.
2014+b=3550
b=3550-2014
b=1536
2014+a=3464
a=3464-2014
a=1450
DOUBLE CHECK
a+b=2986
1450+1536=2986 is true
Therefore, the biggest number is 2014, the smallest number is 1450 and the middle number is 1536.
whoops I was sleepy and spelled "pairs" wrong
Delete
ReplyDelete2986= a+b
3464= c+a
3550 = c+b
2a+2b+2c=2986+3464+3550=10000
10000=2a+2b+3b
a+b+c=5000
5000-2986(a+b)= 2014
c=2014
Info:
ReplyDelete2986 = a + b
3464 = c + a
3550 = c + b
Math =
2a + 2b + 2c = 10000 = 2986 + 3464 + 3550
10000 / 2 = 5000 = a + b + c
5000 - 2986 (a + b) = 2014
c = 2014
a+b=smallest=2986
ReplyDeletec+a=middle number=3464
b+c= the largest=3550
2986(a+b)+3464(a+c)+3550(b+c)=1000
2a+2b+2c= 1000
1000/2= 500
1000/2= a+b+c
500=a+b+c
500-a+b=c
500-2986=c
c=2014
The answer for c is 2014
POTW:
ReplyDeletea + b = 2986
b + c = 3550
c + a = 3464
2986 + 3550 + 3464 = (a+b) + (b+c) + (c+a) = 2a + 2b + 2c = 1000
1000/2 = 5000
500-2986 = 2014
c = 2014
We know that a ＜ b ＜ c, and:
ReplyDeletea + b = 2986 - Equation 1
a + c = 3464 - Equation 2
b + c = 3550 - Equation 3
This means that:
b + (3464 - a) = 3550 - From equations 2 and 3
b - a = 3550 - 3464 = 86
b - a = 86
b = a + 86
Now, using Equation 1
a + (a + 86) = 2986
2a = 2986 - 86
Therefore,
a = 2900 / 2 = 1450
Now, we know that a = 1450
Therefore, b = a + 86 = 1450 + 86 = 1536
We also know that b = 1536
Finally, 1536 + c = 3550, so 3550 - 1536 = 2014
We know that c = 2014
So, 1450 ＜ 1536 ＜ 2014