These four numbers make 25
$begingroup$
The sum of the whole is 25
Large and small are different by 4
The first is 2 times the fourth
Third is equal to first
What are the four numbers?
This is really bugging me can not figure it out, thanks for all your help....
Source: This thing right here
mathematics
$endgroup$
add a comment |
$begingroup$
The sum of the whole is 25
Large and small are different by 4
The first is 2 times the fourth
Third is equal to first
What are the four numbers?
This is really bugging me can not figure it out, thanks for all your help....
Source: This thing right here
mathematics
$endgroup$
2
$begingroup$
Has a correct answer been given? If so, please don't forget to $color{green}{checkmark smalltext{Accept}}$ it :)
$endgroup$
– Rubio♦
Apr 20 '18 at 22:31
$begingroup$
Do you allow non-integer solutions too?
$endgroup$
– smci
May 4 '18 at 7:58
add a comment |
$begingroup$
The sum of the whole is 25
Large and small are different by 4
The first is 2 times the fourth
Third is equal to first
What are the four numbers?
This is really bugging me can not figure it out, thanks for all your help....
Source: This thing right here
mathematics
$endgroup$
The sum of the whole is 25
Large and small are different by 4
The first is 2 times the fourth
Third is equal to first
What are the four numbers?
This is really bugging me can not figure it out, thanks for all your help....
Source: This thing right here
mathematics
mathematics
edited Apr 19 '18 at 4:02
Rubio♦
29.9k566185
29.9k566185
asked Apr 19 '18 at 3:45
Rachel TappRachel Tapp
9313
9313
2
$begingroup$
Has a correct answer been given? If so, please don't forget to $color{green}{checkmark smalltext{Accept}}$ it :)
$endgroup$
– Rubio♦
Apr 20 '18 at 22:31
$begingroup$
Do you allow non-integer solutions too?
$endgroup$
– smci
May 4 '18 at 7:58
add a comment |
2
$begingroup$
Has a correct answer been given? If so, please don't forget to $color{green}{checkmark smalltext{Accept}}$ it :)
$endgroup$
– Rubio♦
Apr 20 '18 at 22:31
$begingroup$
Do you allow non-integer solutions too?
$endgroup$
– smci
May 4 '18 at 7:58
2
2
$begingroup$
Has a correct answer been given? If so, please don't forget to $color{green}{checkmark smalltext{Accept}}$ it :)
$endgroup$
– Rubio♦
Apr 20 '18 at 22:31
$begingroup$
Has a correct answer been given? If so, please don't forget to $color{green}{checkmark smalltext{Accept}}$ it :)
$endgroup$
– Rubio♦
Apr 20 '18 at 22:31
$begingroup$
Do you allow non-integer solutions too?
$endgroup$
– smci
May 4 '18 at 7:58
$begingroup$
Do you allow non-integer solutions too?
$endgroup$
– smci
May 4 '18 at 7:58
add a comment |
8 Answers
8
active
oldest
votes
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could be:
8,5,8,4
fits all conditions:
$4*2 = 8 implies 8-4 = 4 implies 8+5+8+4 = 25 implies 1st=3rd=8$
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6
$begingroup$
This is the only integer solution.
$endgroup$
– Rubio♦
Apr 19 '18 at 3:53
add a comment |
$begingroup$
by putting all conditions except "Large and small are different by 4" as letters below
$2a$,$b$,$2a$,$a$ where sum is $25$
should be these numbers. so we can conclude that
First condition:
$2a$ could be biggest while $b$ could be lowest, so $2a-b=4$
Second condition:
$b$ could be biggest, $b-a=4$ since $2a$ is greater than $a$.
and Last condition:
$2a$ could be biggest and $a$ could be lowest. $a=4$ then.
In other words,
$5a+b=25$ where $2a-b=4$ or $a=4$ or $b-a=4$
so using first condition;
$7a-4=25$ so $a$ is not integer.
or using second condition;
$6a+4=25$ so $a$ is not integer again.
and using the last condition
$a=4$, $2a=8$ and $b=5$
it seems last condition is the only solution we could have.
$endgroup$
add a comment |
$begingroup$
The first is 2 times the fourth
Third is equal to first
We define the numbers as 2a, b, 2a, a
The sum of the whole is 25
We have 2a + b + 2a + a = 25
Large and small are different by 4
We have max(a, b, 2a) - min(a, b, 2a) = 4
Solving the 2 equations...
We got 2 solutions.
8, 5, 8, 4
7, 7.5, 7, 3.5
$endgroup$
add a comment |
$begingroup$
To expand on a process to get to @Preet's answer.
We know that the biggest number must be bigger than 6. Since $25 / 4 = 6.25$
So the smallest number must be greater than 3.
We know that some number is double the other. If the smallest was 5 than the smallest double is 10, $10 - 5 = 5 > 4$ and it can be easily shown that the difference gets bigger as the small number gets bigger.
We now know that 4 must be the smallest, and the biggest is 8.
Since 8 is double of 4 we know that 4 is the fourth, and 8 is the first, because no other doubles can fit between these numbers
Since the third is the same as the first, we end up with 8, _, 8, 4
We have shown that this is the only integer solution. Using the same logic it may be possible to show that there is only an integer solution.
A proof of whether or not a non-integer solution is possible:
We know that for any value of the smallest number greater than 4, it is impossible for one number to be doubled of another.
We know that $2.25$ is a naive lower bound for the smallest as well. Since $25 / 4 = 6.25$ and $6.25 - 4 = 2.25$
So the lowest number is between 2.25 and 4.
2.25 is only possible if all 4 values were 6.25 (an immediate contradiction).
If we refine the equation such that 3 are the same, and the 4th is 4 less:
$4x - 4 = 25$
$x = 29 / 4 = 7.25$
So the largest number is greater than 7.25 so the lowest must be larger than 3.25
The maximum largest number is just under 8 (limit) if one of the two remaining numbers is less than 4. than the other must be greater than 9 to reach 25 as a sum. 9 is impossible so this is a contradiction.
So one of the middle numbers must be double the smallest. (For any number other than 4, the largest will not be double that of the smallest).
Furthermore we know that one number must exist twice, and it cannot be the smallest. If the largest was copied, than the other must be double of the smallest. $x + 2x + z + z = 25$
$3x + z + z = 25$
$z = x + 4$
$5x + 8 = 25$
$5x = 17$
$x = 3.4$
So we found another solution.
3.4, 6.8, 7.4, 7.4
Finally the last case is if the 2 non-extrema are the same. They must also be double the smallest so we get:
$x + 2x + 2x + x+4 = 25$
$6x+4 = 25$
$6x = 21$
$x = 3.5$
so another solution is
7, 7.5, 7, 3.5
$endgroup$
$begingroup$
seems that3.4, 6.8, 7.4, 7.4
is not a valid answer as it cannot fulfillThird is equal to first
andThe first is 2 times the fourth
at the same time
$endgroup$
– wilson
Apr 19 '18 at 6:47
$begingroup$
Oh that is a good point. I didn't realise there was a constraint that the one that is double the smallest must also have a pair.
$endgroup$
– wolfram42
Apr 19 '18 at 14:03
add a comment |
$begingroup$
It is straightforward:
1) $~ x + y + z + w = 25$
2) $~ x - w = 4$
3) $~ x = 2 times w$
4) $~ x = z$
Then
From 2) and 3) we get: $x = w + 4 = 2 times w ~~ implies w = 4$
From there you are done:
$x = 2 times w = 2 times 4 = 8$
$x = z$ therefore $z=8$
$x + y + z + w = 8 + y + 8 + 4 = 25 implies y = 5$
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1
$begingroup$
Not a given x is max and w is min
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– paparazzo
Apr 19 '18 at 20:06
add a comment |
$begingroup$
My solution:
The numbers are $2a,b,2a,a$. The sum is $25$, so $5a+b=25$ means that $b$ is a multiple of $5$. A simple check using condition (2) means $bne0$ as $2a=4$ gives a total of $10$. $b=10$ fails because then $a=3$ and $10-3=7ne4$.
So $b=5, a=4$ and the numbers are $4,5,8,8$.
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add a comment |
$begingroup$
the first number is 8
the second number is 5
the third number is 8
and the fourth number is 4
New contributor
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add a comment |
$begingroup$
What are the four numbers?
- The sum of the whole is 25:
$a + b + c + d = 25$
- Large and small are different by 4:
$x - y = 4$ where $x > y$ and $x$ could be ($a$ or $b$) and y could be ($b$ or $d$)
- The first is 2 times the fourth:
$(a = 2d) ... 2d + b + c + d = 25 => 3d + b + c = 25$
- Third is equal to first:
$(c = a = 2d) ... 2d + b + 2d + d = 25 => 5d + b = 25$
The step 2 is the key to solve the problem. Just replace a, b or d for x and y (excluding c since c = a
)...
You will find 2 "integer" solutions:
Solution 1:
when $x = a = 2d$, and $y = d$
$x - y = 4 => 2d - d = 4 => d = 4$
Replace that in the last equation:
$5d + b = 25 => 5*4 + b = 25 => 20 + b = 25 => b = 5$
Solution: $a = 8, b = 5, c = 8, d = 4$
Solution 2: when x = b
, and y = a = 2d
b - a = 4 => b = 4 + a => same as => b = 4 + 2d
Replace that in the last equation, to find d:
5d + b = 25 => 5d + (4 + 2d) = 25 => 7d = 25 - 4 => 7d = 21 => d = 3
And then in the case equation, to find b:
b = 4 + 2*3 => b = 10
- a = 2d = 2*3 = 6
- b = 10
- c = 2d = 6
- d = 3
I have to remove this 2nd solution since violates the rule #2. 10-3 = 7... in other words, y
can't be a
since a
is not the smallest (a > d
), Thanks @Rubio.
All the possible cases are:
* $a - b = 4$, it is possible $a > b$ (no-integer solution)
* $a - d = 4$, we are sure $a > d$ (integer solution)
* $b - d = 4$, it is possible $b > d$ (no-integer solution)
Note these are NOT a valid cases:
* $b - a = 4$, we are sure $a$ is NOT the smallest, $a > d$ (error for solution 2)
* $d - b = 4$, we are sure $d$ is NOT the largest, $a > d$
* $a - c = 4$, impossible $a = c$
* $d - a = 4$, impossible $a > d$
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$begingroup$
6,10,6,3 violates Line 2 constraint (which would better have been said as largest and smallest, but the intent is pretty clear). And how is any of your answer substantively different from the several that preceded it (other than half of it being wrong)?
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– Rubio♦
Apr 19 '18 at 23:01
1
$begingroup$
Thanks for all the help, made my day better....
$endgroup$
– Rachel Tapp
Apr 20 '18 at 20:12
add a comment |
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8 Answers
8
active
oldest
votes
8 Answers
8
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
could be:
8,5,8,4
fits all conditions:
$4*2 = 8 implies 8-4 = 4 implies 8+5+8+4 = 25 implies 1st=3rd=8$
$endgroup$
6
$begingroup$
This is the only integer solution.
$endgroup$
– Rubio♦
Apr 19 '18 at 3:53
add a comment |
$begingroup$
could be:
8,5,8,4
fits all conditions:
$4*2 = 8 implies 8-4 = 4 implies 8+5+8+4 = 25 implies 1st=3rd=8$
$endgroup$
6
$begingroup$
This is the only integer solution.
$endgroup$
– Rubio♦
Apr 19 '18 at 3:53
add a comment |
$begingroup$
could be:
8,5,8,4
fits all conditions:
$4*2 = 8 implies 8-4 = 4 implies 8+5+8+4 = 25 implies 1st=3rd=8$
$endgroup$
could be:
8,5,8,4
fits all conditions:
$4*2 = 8 implies 8-4 = 4 implies 8+5+8+4 = 25 implies 1st=3rd=8$
edited Apr 19 '18 at 23:30
kraby15
2,3963930
2,3963930
answered Apr 19 '18 at 3:50
PreetPreet
2,1322731
2,1322731
6
$begingroup$
This is the only integer solution.
$endgroup$
– Rubio♦
Apr 19 '18 at 3:53
add a comment |
6
$begingroup$
This is the only integer solution.
$endgroup$
– Rubio♦
Apr 19 '18 at 3:53
6
6
$begingroup$
This is the only integer solution.
$endgroup$
– Rubio♦
Apr 19 '18 at 3:53
$begingroup$
This is the only integer solution.
$endgroup$
– Rubio♦
Apr 19 '18 at 3:53
add a comment |
$begingroup$
by putting all conditions except "Large and small are different by 4" as letters below
$2a$,$b$,$2a$,$a$ where sum is $25$
should be these numbers. so we can conclude that
First condition:
$2a$ could be biggest while $b$ could be lowest, so $2a-b=4$
Second condition:
$b$ could be biggest, $b-a=4$ since $2a$ is greater than $a$.
and Last condition:
$2a$ could be biggest and $a$ could be lowest. $a=4$ then.
In other words,
$5a+b=25$ where $2a-b=4$ or $a=4$ or $b-a=4$
so using first condition;
$7a-4=25$ so $a$ is not integer.
or using second condition;
$6a+4=25$ so $a$ is not integer again.
and using the last condition
$a=4$, $2a=8$ and $b=5$
it seems last condition is the only solution we could have.
$endgroup$
add a comment |
$begingroup$
by putting all conditions except "Large and small are different by 4" as letters below
$2a$,$b$,$2a$,$a$ where sum is $25$
should be these numbers. so we can conclude that
First condition:
$2a$ could be biggest while $b$ could be lowest, so $2a-b=4$
Second condition:
$b$ could be biggest, $b-a=4$ since $2a$ is greater than $a$.
and Last condition:
$2a$ could be biggest and $a$ could be lowest. $a=4$ then.
In other words,
$5a+b=25$ where $2a-b=4$ or $a=4$ or $b-a=4$
so using first condition;
$7a-4=25$ so $a$ is not integer.
or using second condition;
$6a+4=25$ so $a$ is not integer again.
and using the last condition
$a=4$, $2a=8$ and $b=5$
it seems last condition is the only solution we could have.
$endgroup$
add a comment |
$begingroup$
by putting all conditions except "Large and small are different by 4" as letters below
$2a$,$b$,$2a$,$a$ where sum is $25$
should be these numbers. so we can conclude that
First condition:
$2a$ could be biggest while $b$ could be lowest, so $2a-b=4$
Second condition:
$b$ could be biggest, $b-a=4$ since $2a$ is greater than $a$.
and Last condition:
$2a$ could be biggest and $a$ could be lowest. $a=4$ then.
In other words,
$5a+b=25$ where $2a-b=4$ or $a=4$ or $b-a=4$
so using first condition;
$7a-4=25$ so $a$ is not integer.
or using second condition;
$6a+4=25$ so $a$ is not integer again.
and using the last condition
$a=4$, $2a=8$ and $b=5$
it seems last condition is the only solution we could have.
$endgroup$
by putting all conditions except "Large and small are different by 4" as letters below
$2a$,$b$,$2a$,$a$ where sum is $25$
should be these numbers. so we can conclude that
First condition:
$2a$ could be biggest while $b$ could be lowest, so $2a-b=4$
Second condition:
$b$ could be biggest, $b-a=4$ since $2a$ is greater than $a$.
and Last condition:
$2a$ could be biggest and $a$ could be lowest. $a=4$ then.
In other words,
$5a+b=25$ where $2a-b=4$ or $a=4$ or $b-a=4$
so using first condition;
$7a-4=25$ so $a$ is not integer.
or using second condition;
$6a+4=25$ so $a$ is not integer again.
and using the last condition
$a=4$, $2a=8$ and $b=5$
it seems last condition is the only solution we could have.
edited Apr 19 '18 at 6:49
answered Apr 19 '18 at 5:40
OrayOray
16k436156
16k436156
add a comment |
add a comment |
$begingroup$
The first is 2 times the fourth
Third is equal to first
We define the numbers as 2a, b, 2a, a
The sum of the whole is 25
We have 2a + b + 2a + a = 25
Large and small are different by 4
We have max(a, b, 2a) - min(a, b, 2a) = 4
Solving the 2 equations...
We got 2 solutions.
8, 5, 8, 4
7, 7.5, 7, 3.5
$endgroup$
add a comment |
$begingroup$
The first is 2 times the fourth
Third is equal to first
We define the numbers as 2a, b, 2a, a
The sum of the whole is 25
We have 2a + b + 2a + a = 25
Large and small are different by 4
We have max(a, b, 2a) - min(a, b, 2a) = 4
Solving the 2 equations...
We got 2 solutions.
8, 5, 8, 4
7, 7.5, 7, 3.5
$endgroup$
add a comment |
$begingroup$
The first is 2 times the fourth
Third is equal to first
We define the numbers as 2a, b, 2a, a
The sum of the whole is 25
We have 2a + b + 2a + a = 25
Large and small are different by 4
We have max(a, b, 2a) - min(a, b, 2a) = 4
Solving the 2 equations...
We got 2 solutions.
8, 5, 8, 4
7, 7.5, 7, 3.5
$endgroup$
The first is 2 times the fourth
Third is equal to first
We define the numbers as 2a, b, 2a, a
The sum of the whole is 25
We have 2a + b + 2a + a = 25
Large and small are different by 4
We have max(a, b, 2a) - min(a, b, 2a) = 4
Solving the 2 equations...
We got 2 solutions.
8, 5, 8, 4
7, 7.5, 7, 3.5
answered Apr 19 '18 at 7:05
wilsonwilson
1914
1914
add a comment |
add a comment |
$begingroup$
To expand on a process to get to @Preet's answer.
We know that the biggest number must be bigger than 6. Since $25 / 4 = 6.25$
So the smallest number must be greater than 3.
We know that some number is double the other. If the smallest was 5 than the smallest double is 10, $10 - 5 = 5 > 4$ and it can be easily shown that the difference gets bigger as the small number gets bigger.
We now know that 4 must be the smallest, and the biggest is 8.
Since 8 is double of 4 we know that 4 is the fourth, and 8 is the first, because no other doubles can fit between these numbers
Since the third is the same as the first, we end up with 8, _, 8, 4
We have shown that this is the only integer solution. Using the same logic it may be possible to show that there is only an integer solution.
A proof of whether or not a non-integer solution is possible:
We know that for any value of the smallest number greater than 4, it is impossible for one number to be doubled of another.
We know that $2.25$ is a naive lower bound for the smallest as well. Since $25 / 4 = 6.25$ and $6.25 - 4 = 2.25$
So the lowest number is between 2.25 and 4.
2.25 is only possible if all 4 values were 6.25 (an immediate contradiction).
If we refine the equation such that 3 are the same, and the 4th is 4 less:
$4x - 4 = 25$
$x = 29 / 4 = 7.25$
So the largest number is greater than 7.25 so the lowest must be larger than 3.25
The maximum largest number is just under 8 (limit) if one of the two remaining numbers is less than 4. than the other must be greater than 9 to reach 25 as a sum. 9 is impossible so this is a contradiction.
So one of the middle numbers must be double the smallest. (For any number other than 4, the largest will not be double that of the smallest).
Furthermore we know that one number must exist twice, and it cannot be the smallest. If the largest was copied, than the other must be double of the smallest. $x + 2x + z + z = 25$
$3x + z + z = 25$
$z = x + 4$
$5x + 8 = 25$
$5x = 17$
$x = 3.4$
So we found another solution.
3.4, 6.8, 7.4, 7.4
Finally the last case is if the 2 non-extrema are the same. They must also be double the smallest so we get:
$x + 2x + 2x + x+4 = 25$
$6x+4 = 25$
$6x = 21$
$x = 3.5$
so another solution is
7, 7.5, 7, 3.5
$endgroup$
$begingroup$
seems that3.4, 6.8, 7.4, 7.4
is not a valid answer as it cannot fulfillThird is equal to first
andThe first is 2 times the fourth
at the same time
$endgroup$
– wilson
Apr 19 '18 at 6:47
$begingroup$
Oh that is a good point. I didn't realise there was a constraint that the one that is double the smallest must also have a pair.
$endgroup$
– wolfram42
Apr 19 '18 at 14:03
add a comment |
$begingroup$
To expand on a process to get to @Preet's answer.
We know that the biggest number must be bigger than 6. Since $25 / 4 = 6.25$
So the smallest number must be greater than 3.
We know that some number is double the other. If the smallest was 5 than the smallest double is 10, $10 - 5 = 5 > 4$ and it can be easily shown that the difference gets bigger as the small number gets bigger.
We now know that 4 must be the smallest, and the biggest is 8.
Since 8 is double of 4 we know that 4 is the fourth, and 8 is the first, because no other doubles can fit between these numbers
Since the third is the same as the first, we end up with 8, _, 8, 4
We have shown that this is the only integer solution. Using the same logic it may be possible to show that there is only an integer solution.
A proof of whether or not a non-integer solution is possible:
We know that for any value of the smallest number greater than 4, it is impossible for one number to be doubled of another.
We know that $2.25$ is a naive lower bound for the smallest as well. Since $25 / 4 = 6.25$ and $6.25 - 4 = 2.25$
So the lowest number is between 2.25 and 4.
2.25 is only possible if all 4 values were 6.25 (an immediate contradiction).
If we refine the equation such that 3 are the same, and the 4th is 4 less:
$4x - 4 = 25$
$x = 29 / 4 = 7.25$
So the largest number is greater than 7.25 so the lowest must be larger than 3.25
The maximum largest number is just under 8 (limit) if one of the two remaining numbers is less than 4. than the other must be greater than 9 to reach 25 as a sum. 9 is impossible so this is a contradiction.
So one of the middle numbers must be double the smallest. (For any number other than 4, the largest will not be double that of the smallest).
Furthermore we know that one number must exist twice, and it cannot be the smallest. If the largest was copied, than the other must be double of the smallest. $x + 2x + z + z = 25$
$3x + z + z = 25$
$z = x + 4$
$5x + 8 = 25$
$5x = 17$
$x = 3.4$
So we found another solution.
3.4, 6.8, 7.4, 7.4
Finally the last case is if the 2 non-extrema are the same. They must also be double the smallest so we get:
$x + 2x + 2x + x+4 = 25$
$6x+4 = 25$
$6x = 21$
$x = 3.5$
so another solution is
7, 7.5, 7, 3.5
$endgroup$
$begingroup$
seems that3.4, 6.8, 7.4, 7.4
is not a valid answer as it cannot fulfillThird is equal to first
andThe first is 2 times the fourth
at the same time
$endgroup$
– wilson
Apr 19 '18 at 6:47
$begingroup$
Oh that is a good point. I didn't realise there was a constraint that the one that is double the smallest must also have a pair.
$endgroup$
– wolfram42
Apr 19 '18 at 14:03
add a comment |
$begingroup$
To expand on a process to get to @Preet's answer.
We know that the biggest number must be bigger than 6. Since $25 / 4 = 6.25$
So the smallest number must be greater than 3.
We know that some number is double the other. If the smallest was 5 than the smallest double is 10, $10 - 5 = 5 > 4$ and it can be easily shown that the difference gets bigger as the small number gets bigger.
We now know that 4 must be the smallest, and the biggest is 8.
Since 8 is double of 4 we know that 4 is the fourth, and 8 is the first, because no other doubles can fit between these numbers
Since the third is the same as the first, we end up with 8, _, 8, 4
We have shown that this is the only integer solution. Using the same logic it may be possible to show that there is only an integer solution.
A proof of whether or not a non-integer solution is possible:
We know that for any value of the smallest number greater than 4, it is impossible for one number to be doubled of another.
We know that $2.25$ is a naive lower bound for the smallest as well. Since $25 / 4 = 6.25$ and $6.25 - 4 = 2.25$
So the lowest number is between 2.25 and 4.
2.25 is only possible if all 4 values were 6.25 (an immediate contradiction).
If we refine the equation such that 3 are the same, and the 4th is 4 less:
$4x - 4 = 25$
$x = 29 / 4 = 7.25$
So the largest number is greater than 7.25 so the lowest must be larger than 3.25
The maximum largest number is just under 8 (limit) if one of the two remaining numbers is less than 4. than the other must be greater than 9 to reach 25 as a sum. 9 is impossible so this is a contradiction.
So one of the middle numbers must be double the smallest. (For any number other than 4, the largest will not be double that of the smallest).
Furthermore we know that one number must exist twice, and it cannot be the smallest. If the largest was copied, than the other must be double of the smallest. $x + 2x + z + z = 25$
$3x + z + z = 25$
$z = x + 4$
$5x + 8 = 25$
$5x = 17$
$x = 3.4$
So we found another solution.
3.4, 6.8, 7.4, 7.4
Finally the last case is if the 2 non-extrema are the same. They must also be double the smallest so we get:
$x + 2x + 2x + x+4 = 25$
$6x+4 = 25$
$6x = 21$
$x = 3.5$
so another solution is
7, 7.5, 7, 3.5
$endgroup$
To expand on a process to get to @Preet's answer.
We know that the biggest number must be bigger than 6. Since $25 / 4 = 6.25$
So the smallest number must be greater than 3.
We know that some number is double the other. If the smallest was 5 than the smallest double is 10, $10 - 5 = 5 > 4$ and it can be easily shown that the difference gets bigger as the small number gets bigger.
We now know that 4 must be the smallest, and the biggest is 8.
Since 8 is double of 4 we know that 4 is the fourth, and 8 is the first, because no other doubles can fit between these numbers
Since the third is the same as the first, we end up with 8, _, 8, 4
We have shown that this is the only integer solution. Using the same logic it may be possible to show that there is only an integer solution.
A proof of whether or not a non-integer solution is possible:
We know that for any value of the smallest number greater than 4, it is impossible for one number to be doubled of another.
We know that $2.25$ is a naive lower bound for the smallest as well. Since $25 / 4 = 6.25$ and $6.25 - 4 = 2.25$
So the lowest number is between 2.25 and 4.
2.25 is only possible if all 4 values were 6.25 (an immediate contradiction).
If we refine the equation such that 3 are the same, and the 4th is 4 less:
$4x - 4 = 25$
$x = 29 / 4 = 7.25$
So the largest number is greater than 7.25 so the lowest must be larger than 3.25
The maximum largest number is just under 8 (limit) if one of the two remaining numbers is less than 4. than the other must be greater than 9 to reach 25 as a sum. 9 is impossible so this is a contradiction.
So one of the middle numbers must be double the smallest. (For any number other than 4, the largest will not be double that of the smallest).
Furthermore we know that one number must exist twice, and it cannot be the smallest. If the largest was copied, than the other must be double of the smallest. $x + 2x + z + z = 25$
$3x + z + z = 25$
$z = x + 4$
$5x + 8 = 25$
$5x = 17$
$x = 3.4$
So we found another solution.
3.4, 6.8, 7.4, 7.4
Finally the last case is if the 2 non-extrema are the same. They must also be double the smallest so we get:
$x + 2x + 2x + x+4 = 25$
$6x+4 = 25$
$6x = 21$
$x = 3.5$
so another solution is
7, 7.5, 7, 3.5
edited Apr 19 '18 at 23:12
answered Apr 19 '18 at 4:36
wolfram42wolfram42
2,215119
2,215119
$begingroup$
seems that3.4, 6.8, 7.4, 7.4
is not a valid answer as it cannot fulfillThird is equal to first
andThe first is 2 times the fourth
at the same time
$endgroup$
– wilson
Apr 19 '18 at 6:47
$begingroup$
Oh that is a good point. I didn't realise there was a constraint that the one that is double the smallest must also have a pair.
$endgroup$
– wolfram42
Apr 19 '18 at 14:03
add a comment |
$begingroup$
seems that3.4, 6.8, 7.4, 7.4
is not a valid answer as it cannot fulfillThird is equal to first
andThe first is 2 times the fourth
at the same time
$endgroup$
– wilson
Apr 19 '18 at 6:47
$begingroup$
Oh that is a good point. I didn't realise there was a constraint that the one that is double the smallest must also have a pair.
$endgroup$
– wolfram42
Apr 19 '18 at 14:03
$begingroup$
seems that
3.4, 6.8, 7.4, 7.4
is not a valid answer as it cannot fulfill Third is equal to first
and The first is 2 times the fourth
at the same time$endgroup$
– wilson
Apr 19 '18 at 6:47
$begingroup$
seems that
3.4, 6.8, 7.4, 7.4
is not a valid answer as it cannot fulfill Third is equal to first
and The first is 2 times the fourth
at the same time$endgroup$
– wilson
Apr 19 '18 at 6:47
$begingroup$
Oh that is a good point. I didn't realise there was a constraint that the one that is double the smallest must also have a pair.
$endgroup$
– wolfram42
Apr 19 '18 at 14:03
$begingroup$
Oh that is a good point. I didn't realise there was a constraint that the one that is double the smallest must also have a pair.
$endgroup$
– wolfram42
Apr 19 '18 at 14:03
add a comment |
$begingroup$
It is straightforward:
1) $~ x + y + z + w = 25$
2) $~ x - w = 4$
3) $~ x = 2 times w$
4) $~ x = z$
Then
From 2) and 3) we get: $x = w + 4 = 2 times w ~~ implies w = 4$
From there you are done:
$x = 2 times w = 2 times 4 = 8$
$x = z$ therefore $z=8$
$x + y + z + w = 8 + y + 8 + 4 = 25 implies y = 5$
$endgroup$
1
$begingroup$
Not a given x is max and w is min
$endgroup$
– paparazzo
Apr 19 '18 at 20:06
add a comment |
$begingroup$
It is straightforward:
1) $~ x + y + z + w = 25$
2) $~ x - w = 4$
3) $~ x = 2 times w$
4) $~ x = z$
Then
From 2) and 3) we get: $x = w + 4 = 2 times w ~~ implies w = 4$
From there you are done:
$x = 2 times w = 2 times 4 = 8$
$x = z$ therefore $z=8$
$x + y + z + w = 8 + y + 8 + 4 = 25 implies y = 5$
$endgroup$
1
$begingroup$
Not a given x is max and w is min
$endgroup$
– paparazzo
Apr 19 '18 at 20:06
add a comment |
$begingroup$
It is straightforward:
1) $~ x + y + z + w = 25$
2) $~ x - w = 4$
3) $~ x = 2 times w$
4) $~ x = z$
Then
From 2) and 3) we get: $x = w + 4 = 2 times w ~~ implies w = 4$
From there you are done:
$x = 2 times w = 2 times 4 = 8$
$x = z$ therefore $z=8$
$x + y + z + w = 8 + y + 8 + 4 = 25 implies y = 5$
$endgroup$
It is straightforward:
1) $~ x + y + z + w = 25$
2) $~ x - w = 4$
3) $~ x = 2 times w$
4) $~ x = z$
Then
From 2) and 3) we get: $x = w + 4 = 2 times w ~~ implies w = 4$
From there you are done:
$x = 2 times w = 2 times 4 = 8$
$x = z$ therefore $z=8$
$x + y + z + w = 8 + y + 8 + 4 = 25 implies y = 5$
edited Apr 19 '18 at 12:41
Rubio♦
29.9k566185
29.9k566185
answered Apr 19 '18 at 12:26
useruser
1674
1674
1
$begingroup$
Not a given x is max and w is min
$endgroup$
– paparazzo
Apr 19 '18 at 20:06
add a comment |
1
$begingroup$
Not a given x is max and w is min
$endgroup$
– paparazzo
Apr 19 '18 at 20:06
1
1
$begingroup$
Not a given x is max and w is min
$endgroup$
– paparazzo
Apr 19 '18 at 20:06
$begingroup$
Not a given x is max and w is min
$endgroup$
– paparazzo
Apr 19 '18 at 20:06
add a comment |
$begingroup$
My solution:
The numbers are $2a,b,2a,a$. The sum is $25$, so $5a+b=25$ means that $b$ is a multiple of $5$. A simple check using condition (2) means $bne0$ as $2a=4$ gives a total of $10$. $b=10$ fails because then $a=3$ and $10-3=7ne4$.
So $b=5, a=4$ and the numbers are $4,5,8,8$.
$endgroup$
add a comment |
$begingroup$
My solution:
The numbers are $2a,b,2a,a$. The sum is $25$, so $5a+b=25$ means that $b$ is a multiple of $5$. A simple check using condition (2) means $bne0$ as $2a=4$ gives a total of $10$. $b=10$ fails because then $a=3$ and $10-3=7ne4$.
So $b=5, a=4$ and the numbers are $4,5,8,8$.
$endgroup$
add a comment |
$begingroup$
My solution:
The numbers are $2a,b,2a,a$. The sum is $25$, so $5a+b=25$ means that $b$ is a multiple of $5$. A simple check using condition (2) means $bne0$ as $2a=4$ gives a total of $10$. $b=10$ fails because then $a=3$ and $10-3=7ne4$.
So $b=5, a=4$ and the numbers are $4,5,8,8$.
$endgroup$
My solution:
The numbers are $2a,b,2a,a$. The sum is $25$, so $5a+b=25$ means that $b$ is a multiple of $5$. A simple check using condition (2) means $bne0$ as $2a=4$ gives a total of $10$. $b=10$ fails because then $a=3$ and $10-3=7ne4$.
So $b=5, a=4$ and the numbers are $4,5,8,8$.
answered Apr 20 '18 at 11:05
JonMark PerryJonMark Perry
20.2k64098
20.2k64098
add a comment |
add a comment |
$begingroup$
the first number is 8
the second number is 5
the third number is 8
and the fourth number is 4
New contributor
$endgroup$
add a comment |
$begingroup$
the first number is 8
the second number is 5
the third number is 8
and the fourth number is 4
New contributor
$endgroup$
add a comment |
$begingroup$
the first number is 8
the second number is 5
the third number is 8
and the fourth number is 4
New contributor
$endgroup$
the first number is 8
the second number is 5
the third number is 8
and the fourth number is 4
New contributor
edited 55 mins ago
Omega Krypton
4,8052444
4,8052444
New contributor
answered 1 hour ago
Andrew Aponte-riveraAndrew Aponte-rivera
36
36
New contributor
New contributor
add a comment |
add a comment |
$begingroup$
What are the four numbers?
- The sum of the whole is 25:
$a + b + c + d = 25$
- Large and small are different by 4:
$x - y = 4$ where $x > y$ and $x$ could be ($a$ or $b$) and y could be ($b$ or $d$)
- The first is 2 times the fourth:
$(a = 2d) ... 2d + b + c + d = 25 => 3d + b + c = 25$
- Third is equal to first:
$(c = a = 2d) ... 2d + b + 2d + d = 25 => 5d + b = 25$
The step 2 is the key to solve the problem. Just replace a, b or d for x and y (excluding c since c = a
)...
You will find 2 "integer" solutions:
Solution 1:
when $x = a = 2d$, and $y = d$
$x - y = 4 => 2d - d = 4 => d = 4$
Replace that in the last equation:
$5d + b = 25 => 5*4 + b = 25 => 20 + b = 25 => b = 5$
Solution: $a = 8, b = 5, c = 8, d = 4$
Solution 2: when x = b
, and y = a = 2d
b - a = 4 => b = 4 + a => same as => b = 4 + 2d
Replace that in the last equation, to find d:
5d + b = 25 => 5d + (4 + 2d) = 25 => 7d = 25 - 4 => 7d = 21 => d = 3
And then in the case equation, to find b:
b = 4 + 2*3 => b = 10
- a = 2d = 2*3 = 6
- b = 10
- c = 2d = 6
- d = 3
I have to remove this 2nd solution since violates the rule #2. 10-3 = 7... in other words, y
can't be a
since a
is not the smallest (a > d
), Thanks @Rubio.
All the possible cases are:
* $a - b = 4$, it is possible $a > b$ (no-integer solution)
* $a - d = 4$, we are sure $a > d$ (integer solution)
* $b - d = 4$, it is possible $b > d$ (no-integer solution)
Note these are NOT a valid cases:
* $b - a = 4$, we are sure $a$ is NOT the smallest, $a > d$ (error for solution 2)
* $d - b = 4$, we are sure $d$ is NOT the largest, $a > d$
* $a - c = 4$, impossible $a = c$
* $d - a = 4$, impossible $a > d$
$endgroup$
$begingroup$
6,10,6,3 violates Line 2 constraint (which would better have been said as largest and smallest, but the intent is pretty clear). And how is any of your answer substantively different from the several that preceded it (other than half of it being wrong)?
$endgroup$
– Rubio♦
Apr 19 '18 at 23:01
1
$begingroup$
Thanks for all the help, made my day better....
$endgroup$
– Rachel Tapp
Apr 20 '18 at 20:12
add a comment |
$begingroup$
What are the four numbers?
- The sum of the whole is 25:
$a + b + c + d = 25$
- Large and small are different by 4:
$x - y = 4$ where $x > y$ and $x$ could be ($a$ or $b$) and y could be ($b$ or $d$)
- The first is 2 times the fourth:
$(a = 2d) ... 2d + b + c + d = 25 => 3d + b + c = 25$
- Third is equal to first:
$(c = a = 2d) ... 2d + b + 2d + d = 25 => 5d + b = 25$
The step 2 is the key to solve the problem. Just replace a, b or d for x and y (excluding c since c = a
)...
You will find 2 "integer" solutions:
Solution 1:
when $x = a = 2d$, and $y = d$
$x - y = 4 => 2d - d = 4 => d = 4$
Replace that in the last equation:
$5d + b = 25 => 5*4 + b = 25 => 20 + b = 25 => b = 5$
Solution: $a = 8, b = 5, c = 8, d = 4$
Solution 2: when x = b
, and y = a = 2d
b - a = 4 => b = 4 + a => same as => b = 4 + 2d
Replace that in the last equation, to find d:
5d + b = 25 => 5d + (4 + 2d) = 25 => 7d = 25 - 4 => 7d = 21 => d = 3
And then in the case equation, to find b:
b = 4 + 2*3 => b = 10
- a = 2d = 2*3 = 6
- b = 10
- c = 2d = 6
- d = 3
I have to remove this 2nd solution since violates the rule #2. 10-3 = 7... in other words, y
can't be a
since a
is not the smallest (a > d
), Thanks @Rubio.
All the possible cases are:
* $a - b = 4$, it is possible $a > b$ (no-integer solution)
* $a - d = 4$, we are sure $a > d$ (integer solution)
* $b - d = 4$, it is possible $b > d$ (no-integer solution)
Note these are NOT a valid cases:
* $b - a = 4$, we are sure $a$ is NOT the smallest, $a > d$ (error for solution 2)
* $d - b = 4$, we are sure $d$ is NOT the largest, $a > d$
* $a - c = 4$, impossible $a = c$
* $d - a = 4$, impossible $a > d$
$endgroup$
$begingroup$
6,10,6,3 violates Line 2 constraint (which would better have been said as largest and smallest, but the intent is pretty clear). And how is any of your answer substantively different from the several that preceded it (other than half of it being wrong)?
$endgroup$
– Rubio♦
Apr 19 '18 at 23:01
1
$begingroup$
Thanks for all the help, made my day better....
$endgroup$
– Rachel Tapp
Apr 20 '18 at 20:12
add a comment |
$begingroup$
What are the four numbers?
- The sum of the whole is 25:
$a + b + c + d = 25$
- Large and small are different by 4:
$x - y = 4$ where $x > y$ and $x$ could be ($a$ or $b$) and y could be ($b$ or $d$)
- The first is 2 times the fourth:
$(a = 2d) ... 2d + b + c + d = 25 => 3d + b + c = 25$
- Third is equal to first:
$(c = a = 2d) ... 2d + b + 2d + d = 25 => 5d + b = 25$
The step 2 is the key to solve the problem. Just replace a, b or d for x and y (excluding c since c = a
)...
You will find 2 "integer" solutions:
Solution 1:
when $x = a = 2d$, and $y = d$
$x - y = 4 => 2d - d = 4 => d = 4$
Replace that in the last equation:
$5d + b = 25 => 5*4 + b = 25 => 20 + b = 25 => b = 5$
Solution: $a = 8, b = 5, c = 8, d = 4$
Solution 2: when x = b
, and y = a = 2d
b - a = 4 => b = 4 + a => same as => b = 4 + 2d
Replace that in the last equation, to find d:
5d + b = 25 => 5d + (4 + 2d) = 25 => 7d = 25 - 4 => 7d = 21 => d = 3
And then in the case equation, to find b:
b = 4 + 2*3 => b = 10
- a = 2d = 2*3 = 6
- b = 10
- c = 2d = 6
- d = 3
I have to remove this 2nd solution since violates the rule #2. 10-3 = 7... in other words, y
can't be a
since a
is not the smallest (a > d
), Thanks @Rubio.
All the possible cases are:
* $a - b = 4$, it is possible $a > b$ (no-integer solution)
* $a - d = 4$, we are sure $a > d$ (integer solution)
* $b - d = 4$, it is possible $b > d$ (no-integer solution)
Note these are NOT a valid cases:
* $b - a = 4$, we are sure $a$ is NOT the smallest, $a > d$ (error for solution 2)
* $d - b = 4$, we are sure $d$ is NOT the largest, $a > d$
* $a - c = 4$, impossible $a = c$
* $d - a = 4$, impossible $a > d$
$endgroup$
What are the four numbers?
- The sum of the whole is 25:
$a + b + c + d = 25$
- Large and small are different by 4:
$x - y = 4$ where $x > y$ and $x$ could be ($a$ or $b$) and y could be ($b$ or $d$)
- The first is 2 times the fourth:
$(a = 2d) ... 2d + b + c + d = 25 => 3d + b + c = 25$
- Third is equal to first:
$(c = a = 2d) ... 2d + b + 2d + d = 25 => 5d + b = 25$
The step 2 is the key to solve the problem. Just replace a, b or d for x and y (excluding c since c = a
)...
You will find 2 "integer" solutions:
Solution 1:
when $x = a = 2d$, and $y = d$
$x - y = 4 => 2d - d = 4 => d = 4$
Replace that in the last equation:
$5d + b = 25 => 5*4 + b = 25 => 20 + b = 25 => b = 5$
Solution: $a = 8, b = 5, c = 8, d = 4$
Solution 2: when x = b
, and y = a = 2d
b - a = 4 => b = 4 + a => same as => b = 4 + 2d
Replace that in the last equation, to find d:
5d + b = 25 => 5d + (4 + 2d) = 25 => 7d = 25 - 4 => 7d = 21 => d = 3
And then in the case equation, to find b:
b = 4 + 2*3 => b = 10
- a = 2d = 2*3 = 6
- b = 10
- c = 2d = 6
- d = 3
I have to remove this 2nd solution since violates the rule #2. 10-3 = 7... in other words, y
can't be a
since a
is not the smallest (a > d
), Thanks @Rubio.
All the possible cases are:
* $a - b = 4$, it is possible $a > b$ (no-integer solution)
* $a - d = 4$, we are sure $a > d$ (integer solution)
* $b - d = 4$, it is possible $b > d$ (no-integer solution)
Note these are NOT a valid cases:
* $b - a = 4$, we are sure $a$ is NOT the smallest, $a > d$ (error for solution 2)
* $d - b = 4$, we are sure $d$ is NOT the largest, $a > d$
* $a - c = 4$, impossible $a = c$
* $d - a = 4$, impossible $a > d$
edited Jun 8 '18 at 22:10
answered Apr 19 '18 at 21:53
JaiderJaider
1073
1073
$begingroup$
6,10,6,3 violates Line 2 constraint (which would better have been said as largest and smallest, but the intent is pretty clear). And how is any of your answer substantively different from the several that preceded it (other than half of it being wrong)?
$endgroup$
– Rubio♦
Apr 19 '18 at 23:01
1
$begingroup$
Thanks for all the help, made my day better....
$endgroup$
– Rachel Tapp
Apr 20 '18 at 20:12
add a comment |
$begingroup$
6,10,6,3 violates Line 2 constraint (which would better have been said as largest and smallest, but the intent is pretty clear). And how is any of your answer substantively different from the several that preceded it (other than half of it being wrong)?
$endgroup$
– Rubio♦
Apr 19 '18 at 23:01
1
$begingroup$
Thanks for all the help, made my day better....
$endgroup$
– Rachel Tapp
Apr 20 '18 at 20:12
$begingroup$
6,10,6,3 violates Line 2 constraint (which would better have been said as largest and smallest, but the intent is pretty clear). And how is any of your answer substantively different from the several that preceded it (other than half of it being wrong)?
$endgroup$
– Rubio♦
Apr 19 '18 at 23:01
$begingroup$
6,10,6,3 violates Line 2 constraint (which would better have been said as largest and smallest, but the intent is pretty clear). And how is any of your answer substantively different from the several that preceded it (other than half of it being wrong)?
$endgroup$
– Rubio♦
Apr 19 '18 at 23:01
1
1
$begingroup$
Thanks for all the help, made my day better....
$endgroup$
– Rachel Tapp
Apr 20 '18 at 20:12
$begingroup$
Thanks for all the help, made my day better....
$endgroup$
– Rachel Tapp
Apr 20 '18 at 20:12
add a comment |
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2
$begingroup$
Has a correct answer been given? If so, please don't forget to $color{green}{checkmark smalltext{Accept}}$ it :)
$endgroup$
– Rubio♦
Apr 20 '18 at 22:31
$begingroup$
Do you allow non-integer solutions too?
$endgroup$
– smci
May 4 '18 at 7:58