Mixing
Toy Soldiers Array
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I have a large number of toy soldiers, which I can arrange into a rectangular array consisting of a number of rows and a number of columns. I notice that if I remove 100 toy soldiers, then I can arrange the remaining ones into a rectangular array with 5 fewer rows and 5 more columns. How many toy soldiers would I have to remove from the original configuration to be able to arrange the remaining ones into a rectangular array with 11 fewer rows and 11 more columns?
This is one of the last questions in a Mathematics Competition that I attempted. I honestly have no clue where to start when solving this question, I asked several maths teachers at my school, but they still haven't managed to find an answer. Any help with this question is appreciated.
Edit:
Since I was asked to show how I attempted to solve this question, here it is.
x = number of soldiers,
r = rows,
c = columns
In the original configuration, the statement x = r * c is true. In the second configuration, the statement x - 100 = (r - 5) * (c + 5) is true. This statement simplified is x = cr + 5r - 5c + 75. In both of these cases, I have three unknown variables. I don't know where to go from here.
algebra-precalculus problem-solving
edited Aug 3 at 12:44
Batominovski
22.6k22776
asked Aug 3 at 11:50
skillz21
33
 |Â
show 4 more comments
up vote
0
down vote
favorite
I have a large number of toy soldiers, which I can arrange into a rectangular array consisting of a number of rows and a number of columns. I notice that if I remove 100 toy soldiers, then I can arrange the remaining ones into a rectangular array with 5 fewer rows and 5 more columns. How many toy soldiers would I have to remove from the original configuration to be able to arrange the remaining ones into a rectangular array with 11 fewer rows and 11 more columns?
This is one of the last questions in a Mathematics Competition that I attempted. I honestly have no clue where to start when solving this question, I asked several maths teachers at my school, but they still haven't managed to find an answer. Any help with this question is appreciated.
Edit:
Since I was asked to show how I attempted to solve this question, here it is.
x = number of soldiers,
r = rows,
c = columns
In the original configuration, the statement x = r * c is true. In the second configuration, the statement x - 100 = (r - 5) * (c + 5) is true. This statement simplified is x = cr + 5r - 5c + 75. In both of these cases, I have three unknown variables. I don't know where to go from here.
algebra-precalculus problem-solving
edited Aug 3 at 12:44
Batominovski
22.6k22776
asked Aug 3 at 11:50
skillz21
33
Express the statements as algebraic equations, with unknowns, and simply solve them.
– David G. Stork
Aug 3 at 11:54
@David G. Stork Any other pointers? I mean, my maths teacher had to ask someone else to solve this problem, if it is as simple as you say, surely he could have done it?
– skillz21
Aug 3 at 11:56
Show your attempts in your problem, and you're far more likely to get help.
– David G. Stork
Aug 3 at 11:58
@David G. Stork That's the problem, I don't know where to start.
– skillz21
Aug 3 at 11:58
Do you have the answer?
– Meeta Jo
Aug 3 at 12:19
 |Â
show 4 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have a large number of toy soldiers, which I can arrange into a rectangular array consisting of a number of rows and a number of columns. I notice that if I remove 100 toy soldiers, then I can arrange the remaining ones into a rectangular array with 5 fewer rows and 5 more columns. How many toy soldiers would I have to remove from the original configuration to be able to arrange the remaining ones into a rectangular array with 11 fewer rows and 11 more columns?
This is one of the last questions in a Mathematics Competition that I attempted. I honestly have no clue where to start when solving this question, I asked several maths teachers at my school, but they still haven't managed to find an answer. Any help with this question is appreciated.
Edit:
Since I was asked to show how I attempted to solve this question, here it is.
x = number of soldiers,
r = rows,
c = columns
In the original configuration, the statement x = r * c is true. In the second configuration, the statement x - 100 = (r - 5) * (c + 5) is true. This statement simplified is x = cr + 5r - 5c + 75. In both of these cases, I have three unknown variables. I don't know where to go from here.
algebra-precalculus problem-solving
edited Aug 3 at 12:44
Batominovski
22.6k22776
asked Aug 3 at 11:50
skillz21
33
I have a large number of toy soldiers, which I can arrange into a rectangular array consisting of a number of rows and a number of columns. I notice that if I remove 100 toy soldiers, then I can arrange the remaining ones into a rectangular array with 5 fewer rows and 5 more columns. How many toy soldiers would I have to remove from the original configuration to be able to arrange the remaining ones into a rectangular array with 11 fewer rows and 11 more columns?
This is one of the last questions in a Mathematics Competition that I attempted. I honestly have no clue where to start when solving this question, I asked several maths teachers at my school, but they still haven't managed to find an answer. Any help with this question is appreciated.
Edit:
Since I was asked to show how I attempted to solve this question, here it is.
x = number of soldiers,
r = rows,
c = columns
In the original configuration, the statement x = r * c is true. In the second configuration, the statement x - 100 = (r - 5) * (c + 5) is true. This statement simplified is x = cr + 5r - 5c + 75. In both of these cases, I have three unknown variables. I don't know where to go from here.
algebra-precalculus problem-solving
edited Aug 3 at 12:44
Batominovski
22.6k22776
asked Aug 3 at 11:50
skillz21
33
edited Aug 3 at 12:44
Batominovski
22.6k22776
edited Aug 3 at 12:44
Batominovski
22.6k22776
edited Aug 3 at 12:44
Batominovski
22.6k22776
22.6k22776
asked Aug 3 at 11:50
skillz21
33
asked Aug 3 at 11:50
skillz21
33
asked Aug 3 at 11:50
skillz21
33
33
Express the statements as algebraic equations, with unknowns, and simply solve them.
– David G. Stork
Aug 3 at 11:54
@David G. Stork Any other pointers? I mean, my maths teacher had to ask someone else to solve this problem, if it is as simple as you say, surely he could have done it?
– skillz21
Aug 3 at 11:56
Show your attempts in your problem, and you're far more likely to get help.
– David G. Stork
Aug 3 at 11:58
@David G. Stork That's the problem, I don't know where to start.
– skillz21
Aug 3 at 11:58
Do you have the answer?
– Meeta Jo
Aug 3 at 12:19
 |Â
show 4 more comments
Express the statements as algebraic equations, with unknowns, and simply solve them.
– David G. Stork
Aug 3 at 11:54
@David G. Stork Any other pointers? I mean, my maths teacher had to ask someone else to solve this problem, if it is as simple as you say, surely he could have done it?
– skillz21
Aug 3 at 11:56
Show your attempts in your problem, and you're far more likely to get help.
– David G. Stork
Aug 3 at 11:58
@David G. Stork That's the problem, I don't know where to start.
– skillz21
Aug 3 at 11:58
Do you have the answer?
– Meeta Jo
Aug 3 at 12:19
Express the statements as algebraic equations, with unknowns, and simply solve them.
– David G. Stork
Aug 3 at 11:54
Express the statements as algebraic equations, with unknowns, and simply solve them.
– David G. Stork
Aug 3 at 11:54
@David G. Stork Any other pointers? I mean, my maths teacher had to ask someone else to solve this problem, if it is as simple as you say, surely he could have done it?
– skillz21
Aug 3 at 11:56
@David G. Stork Any other pointers? I mean, my maths teacher had to ask someone else to solve this problem, if it is as simple as you say, surely he could have done it?
– skillz21
Aug 3 at 11:56
Show your attempts in your problem, and you're far more likely to get help.
– David G. Stork
Aug 3 at 11:58
Show your attempts in your problem, and you're far more likely to get help.
– David G. Stork
Aug 3 at 11:58
@David G. Stork That's the problem, I don't know where to start.
– skillz21
Aug 3 at 11:58
@David G. Stork That's the problem, I don't know where to start.
– skillz21
Aug 3 at 11:58
Do you have the answer?
– Meeta Jo
Aug 3 at 12:19
Do you have the answer?
– Meeta Jo
Aug 3 at 12:19
 |Â
show 4 more comments
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
You have a sign wrong in one equation, so you should actually have the equations
$$n=rc\
n-100=(r-5)(c+5)$$
where $r$, $c$ are the number of rows and columns, and $n$ is the total number of soldiers. Substituting $n$ gives:
$$rc-100=(r-5)(c+5)\
rc-100=rc-5c+5r-25\
c-r=15$$
What the question asks for is how many soldiers are removed when you reduce the number of rows by $11$ and increase the number of columns by $11$. So you want to know:
$$n-(r-11)(c+11) \
= rc-(rc-11c+11r-121)\
= 11c-11r+121\
= 11(c-r)+121$$
But we already know $c-r=15$, so you need to remove $11*15+121 = 286$.
answered Aug 3 at 12:30
Jaap Scherphuis
3,023213
The answer is 286... Not 154.
– skillz21
Aug 3 at 12:38
This is exactly right, but everywhere it says 'c-5' and 'c-11' it should be 'c+5' and 'c+11'. I got 286 following this method with the changes.
– Meeta Jo
Aug 3 at 12:42
Fixed it now. I started with the given equation and didn't see it was incorrect.
– Jaap Scherphuis
Aug 3 at 12:45
I'm sorry, I didn't see that I made a mistake.
– skillz21
Aug 3 at 12:45
add a comment |Â
up vote
1
down vote
From this quote "I notice that if I remove $100$ toy soldiers, then I can arrange the remaining ones into a rectangular array with $5$ fewer rows and $5$ more columns," the correct equation is $$rc-100=(r-5)(c+5),.$$
That is, $$r-c=-15,.$$
Thus,
$$(r-11)(c+11)=rc+11(r-c)-121=rc-11cdot 15-121=rc-286,.$$ Therefore, $286$ toys must be removed.
answered Aug 3 at 12:43
Batominovski
22.6k22776
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
You have a sign wrong in one equation, so you should actually have the equations
$$n=rc\
n-100=(r-5)(c+5)$$
where $r$, $c$ are the number of rows and columns, and $n$ is the total number of soldiers. Substituting $n$ gives:
$$rc-100=(r-5)(c+5)\
rc-100=rc-5c+5r-25\
c-r=15$$
What the question asks for is how many soldiers are removed when you reduce the number of rows by $11$ and increase the number of columns by $11$. So you want to know:
$$n-(r-11)(c+11) \
= rc-(rc-11c+11r-121)\
= 11c-11r+121\
= 11(c-r)+121$$
But we already know $c-r=15$, so you need to remove $11*15+121 = 286$.
answered Aug 3 at 12:30
Jaap Scherphuis
3,023213
The answer is 286... Not 154.
– skillz21
Aug 3 at 12:38
This is exactly right, but everywhere it says 'c-5' and 'c-11' it should be 'c+5' and 'c+11'. I got 286 following this method with the changes.
– Meeta Jo
Aug 3 at 12:42
Fixed it now. I started with the given equation and didn't see it was incorrect.
– Jaap Scherphuis
Aug 3 at 12:45
I'm sorry, I didn't see that I made a mistake.
– skillz21
Aug 3 at 12:45
add a comment |Â
up vote
1
down vote
accepted
You have a sign wrong in one equation, so you should actually have the equations
$$n=rc\
n-100=(r-5)(c+5)$$
where $r$, $c$ are the number of rows and columns, and $n$ is the total number of soldiers. Substituting $n$ gives:
$$rc-100=(r-5)(c+5)\
rc-100=rc-5c+5r-25\
c-r=15$$
What the question asks for is how many soldiers are removed when you reduce the number of rows by $11$ and increase the number of columns by $11$. So you want to know:
$$n-(r-11)(c+11) \
= rc-(rc-11c+11r-121)\
= 11c-11r+121\
= 11(c-r)+121$$
But we already know $c-r=15$, so you need to remove $11*15+121 = 286$.
answered Aug 3 at 12:30
Jaap Scherphuis
3,023213
The answer is 286... Not 154.
– skillz21
Aug 3 at 12:38
This is exactly right, but everywhere it says 'c-5' and 'c-11' it should be 'c+5' and 'c+11'. I got 286 following this method with the changes.
– Meeta Jo
Aug 3 at 12:42
Fixed it now. I started with the given equation and didn't see it was incorrect.
– Jaap Scherphuis
Aug 3 at 12:45
I'm sorry, I didn't see that I made a mistake.
– skillz21
Aug 3 at 12:45
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
You have a sign wrong in one equation, so you should actually have the equations
$$n=rc\
n-100=(r-5)(c+5)$$
where $r$, $c$ are the number of rows and columns, and $n$ is the total number of soldiers. Substituting $n$ gives:
$$rc-100=(r-5)(c+5)\
rc-100=rc-5c+5r-25\
c-r=15$$
What the question asks for is how many soldiers are removed when you reduce the number of rows by $11$ and increase the number of columns by $11$. So you want to know:
$$n-(r-11)(c+11) \
= rc-(rc-11c+11r-121)\
= 11c-11r+121\
= 11(c-r)+121$$
But we already know $c-r=15$, so you need to remove $11*15+121 = 286$.
answered Aug 3 at 12:30
Jaap Scherphuis
3,023213
You have a sign wrong in one equation, so you should actually have the equations
$$n=rc\
n-100=(r-5)(c+5)$$
where $r$, $c$ are the number of rows and columns, and $n$ is the total number of soldiers. Substituting $n$ gives:
$$rc-100=(r-5)(c+5)\
rc-100=rc-5c+5r-25\
c-r=15$$
What the question asks for is how many soldiers are removed when you reduce the number of rows by $11$ and increase the number of columns by $11$. So you want to know:
$$n-(r-11)(c+11) \
= rc-(rc-11c+11r-121)\
= 11c-11r+121\
= 11(c-r)+121$$
But we already know $c-r=15$, so you need to remove $11*15+121 = 286$.
answered Aug 3 at 12:30
Jaap Scherphuis
3,023213
edited Aug 3 at 12:45
answered Aug 3 at 12:30
Jaap Scherphuis
3,023213
answered Aug 3 at 12:30
Jaap Scherphuis
3,023213
answered Aug 3 at 12:30
Jaap Scherphuis
3,023213
3,023213
The answer is 286... Not 154.
– skillz21
Aug 3 at 12:38
This is exactly right, but everywhere it says 'c-5' and 'c-11' it should be 'c+5' and 'c+11'. I got 286 following this method with the changes.
– Meeta Jo
Aug 3 at 12:42
Fixed it now. I started with the given equation and didn't see it was incorrect.
– Jaap Scherphuis
Aug 3 at 12:45
I'm sorry, I didn't see that I made a mistake.
– skillz21
Aug 3 at 12:45
add a comment |Â
The answer is 286... Not 154.
– skillz21
Aug 3 at 12:38
This is exactly right, but everywhere it says 'c-5' and 'c-11' it should be 'c+5' and 'c+11'. I got 286 following this method with the changes.
– Meeta Jo
Aug 3 at 12:42
Fixed it now. I started with the given equation and didn't see it was incorrect.
– Jaap Scherphuis
Aug 3 at 12:45
I'm sorry, I didn't see that I made a mistake.
– skillz21
Aug 3 at 12:45
The answer is 286... Not 154.
– skillz21
Aug 3 at 12:38
The answer is 286... Not 154.
– skillz21
Aug 3 at 12:38
This is exactly right, but everywhere it says 'c-5' and 'c-11' it should be 'c+5' and 'c+11'. I got 286 following this method with the changes.
– Meeta Jo
Aug 3 at 12:42
This is exactly right, but everywhere it says 'c-5' and 'c-11' it should be 'c+5' and 'c+11'. I got 286 following this method with the changes.
– Meeta Jo
Aug 3 at 12:42
Fixed it now. I started with the given equation and didn't see it was incorrect.
– Jaap Scherphuis
Aug 3 at 12:45
Fixed it now. I started with the given equation and didn't see it was incorrect.
– Jaap Scherphuis
Aug 3 at 12:45
I'm sorry, I didn't see that I made a mistake.
– skillz21
Aug 3 at 12:45
I'm sorry, I didn't see that I made a mistake.
– skillz21
Aug 3 at 12:45
add a comment |Â
up vote
1
down vote
From this quote "I notice that if I remove $100$ toy soldiers, then I can arrange the remaining ones into a rectangular array with $5$ fewer rows and $5$ more columns," the correct equation is $$rc-100=(r-5)(c+5),.$$
That is, $$r-c=-15,.$$
Thus,
$$(r-11)(c+11)=rc+11(r-c)-121=rc-11cdot 15-121=rc-286,.$$ Therefore, $286$ toys must be removed.
answered Aug 3 at 12:43
Batominovski
22.6k22776
add a comment |Â
up vote
1
down vote
From this quote "I notice that if I remove $100$ toy soldiers, then I can arrange the remaining ones into a rectangular array with $5$ fewer rows and $5$ more columns," the correct equation is $$rc-100=(r-5)(c+5),.$$
That is, $$r-c=-15,.$$
Thus,
$$(r-11)(c+11)=rc+11(r-c)-121=rc-11cdot 15-121=rc-286,.$$ Therefore, $286$ toys must be removed.
answered Aug 3 at 12:43
Batominovski
22.6k22776
add a comment |Â
up vote
1
down vote
up vote
1
down vote
From this quote "I notice that if I remove $100$ toy soldiers, then I can arrange the remaining ones into a rectangular array with $5$ fewer rows and $5$ more columns," the correct equation is $$rc-100=(r-5)(c+5),.$$
That is, $$r-c=-15,.$$
Thus,
$$(r-11)(c+11)=rc+11(r-c)-121=rc-11cdot 15-121=rc-286,.$$ Therefore, $286$ toys must be removed.
answered Aug 3 at 12:43
Batominovski
22.6k22776
From this quote "I notice that if I remove $100$ toy soldiers, then I can arrange the remaining ones into a rectangular array with $5$ fewer rows and $5$ more columns," the correct equation is $$rc-100=(r-5)(c+5),.$$
That is, $$r-c=-15,.$$
Thus,
$$(r-11)(c+11)=rc+11(r-c)-121=rc-11cdot 15-121=rc-286,.$$ Therefore, $286$ toys must be removed.
answered Aug 3 at 12:43
Batominovski
22.6k22776
edited Aug 3 at 12:50
answered Aug 3 at 12:43
Batominovski
22.6k22776
answered Aug 3 at 12:43
Batominovski
22.6k22776
answered Aug 3 at 12:43
Batominovski
22.6k22776
22.6k22776
add a comment |Â
add a comment |Â
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Express the statements as algebraic equations, with unknowns, and simply solve them.
– David G. Stork
Aug 3 at 11:54
@David G. Stork Any other pointers? I mean, my maths teacher had to ask someone else to solve this problem, if it is as simple as you say, surely he could have done it?
– skillz21
Aug 3 at 11:56
Show your attempts in your problem, and you're far more likely to get help.
– David G. Stork
Aug 3 at 11:58
@David G. Stork That's the problem, I don't know where to start.
– skillz21
Aug 3 at 11:58
Do you have the answer?
– Meeta Jo
Aug 3 at 12:19