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A dollar amount $$-67.9-$ is divisible by 72. What is this number?
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A client buys 72 turkeys. On the receipt, 2 digits are missing, the first and the last. So, the total price is $$-67.9-$. Find a way to make the price of each turkey round at the second digit without trial-and-error method.
So, the answer is obviously $$5.11$/turkey, so $$367.92$ total. I found that with with 2-3 trial-and-error.
I bypassed his problem by using python and made a script that tries every combination and assures it is divisible by 0.01.
Please help me find a mathematical way of solving this problem.
linear-algebra computational-mathematics rounding-error
edited yesterday
Xander Henderson
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asked yesterday
PyThagoras
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up vote
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A client buys 72 turkeys. On the receipt, 2 digits are missing, the first and the last. So, the total price is $$-67.9-$. Find a way to make the price of each turkey round at the second digit without trial-and-error method.
So, the answer is obviously $$5.11$/turkey, so $$367.92$ total. I found that with with 2-3 trial-and-error.
I bypassed his problem by using python and made a script that tries every combination and assures it is divisible by 0.01.
Please help me find a mathematical way of solving this problem.
linear-algebra computational-mathematics rounding-error
edited yesterday
Xander Henderson
13.3k83250
asked yesterday
PyThagoras
285
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A client buys 72 turkeys. On the receipt, 2 digits are missing, the first and the last. So, the total price is $$-67.9-$. Find a way to make the price of each turkey round at the second digit without trial-and-error method.
So, the answer is obviously $$5.11$/turkey, so $$367.92$ total. I found that with with 2-3 trial-and-error.
I bypassed his problem by using python and made a script that tries every combination and assures it is divisible by 0.01.
Please help me find a mathematical way of solving this problem.
linear-algebra computational-mathematics rounding-error
edited yesterday
Xander Henderson
13.3k83250
asked yesterday
PyThagoras
285
A client buys 72 turkeys. On the receipt, 2 digits are missing, the first and the last. So, the total price is $$-67.9-$. Find a way to make the price of each turkey round at the second digit without trial-and-error method.
So, the answer is obviously $$5.11$/turkey, so $$367.92$ total. I found that with with 2-3 trial-and-error.
I bypassed his problem by using python and made a script that tries every combination and assures it is divisible by 0.01.
Please help me find a mathematical way of solving this problem.
linear-algebra computational-mathematics rounding-error
linear-algebra computational-mathematics rounding-error
edited yesterday
Xander Henderson
13.3k83250
asked yesterday
PyThagoras
285
edited yesterday
Xander Henderson
13.3k83250
asked yesterday
PyThagoras
285
edited yesterday
Xander Henderson
13.3k83250
edited yesterday
Xander Henderson
13.3k83250
edited yesterday
Xander Henderson
13.3k83250
13.3k83250
asked yesterday
PyThagoras
285
asked yesterday
PyThagoras
285
asked yesterday
PyThagoras
285
285
add a comment |Â
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
6
down vote
accepted
You need the divisibility rules for $8,9$ because a number is divisible by $72$ only if it is divisible by $8$ and $9$ as well.
According to that the rule, the sum of the digits must be divisible by $9$.
$$x+6+7+y = 22+x+y$$
In order to be divisible by 9, that sum must either be 27 or 36.
The blanks must sum to $5$ or $14$.
It must also be divisible by $8$ so the last digit must be $2$. That means the first digit has to be $3$, because it would be impossible to make a sum of $14$.
Hence, the answer is x=3 and y=2
edited yesterday
PyThagoras
285
answered yesterday
Ross Millikan
279k23189355
add a comment |Â
up vote
2
down vote
Without loss of generality, we can multiply the prices by 100, to work with integers and avoid the commas.
Let $n$ be the number of turkeys bought and $p$ be the price of each of them.
Then $$72p=10000a+679*10+b$$ where $a,b$ are between $0$ and $9$.
The trick is to notice that $72=9*8$. We can now use two divisibility tricks.
If a number is divisible by $8$, then the last 3 digits must be divisible by it (this follows by the fact that $1000$ is divisible by $8$). Hence $79*10+b$ must be divisible by $8$.
The only value of $b$ which satisfies this condition is $2$.
The second trick concerns divisibility by $9$: if a number is divisible by $9$, then the sum of the digits is divisible by $9$. Hence we must have $$a+6+7+9+2=24+a$$ to be divisible by 9.
The only value of $a$ which satisfies this condition is $3$.
To find the price, we calculate:
$$p=36792/72=511$$
Dividing back by 100, we get $p=5.11$ as you said.
answered yesterday
A-B-izi
12718
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
accepted
You need the divisibility rules for $8,9$ because a number is divisible by $72$ only if it is divisible by $8$ and $9$ as well.
According to that the rule, the sum of the digits must be divisible by $9$.
$$x+6+7+y = 22+x+y$$
In order to be divisible by 9, that sum must either be 27 or 36.
The blanks must sum to $5$ or $14$.
It must also be divisible by $8$ so the last digit must be $2$. That means the first digit has to be $3$, because it would be impossible to make a sum of $14$.
Hence, the answer is x=3 and y=2
edited yesterday
PyThagoras
285
answered yesterday
Ross Millikan
279k23189355
add a comment |Â
up vote
6
down vote
accepted
You need the divisibility rules for $8,9$ because a number is divisible by $72$ only if it is divisible by $8$ and $9$ as well.
According to that the rule, the sum of the digits must be divisible by $9$.
$$x+6+7+y = 22+x+y$$
In order to be divisible by 9, that sum must either be 27 or 36.
The blanks must sum to $5$ or $14$.
It must also be divisible by $8$ so the last digit must be $2$. That means the first digit has to be $3$, because it would be impossible to make a sum of $14$.
Hence, the answer is x=3 and y=2
edited yesterday
PyThagoras
285
answered yesterday
Ross Millikan
279k23189355
add a comment |Â
up vote
6
down vote
accepted
up vote
6
down vote
accepted
You need the divisibility rules for $8,9$ because a number is divisible by $72$ only if it is divisible by $8$ and $9$ as well.
According to that the rule, the sum of the digits must be divisible by $9$.
$$x+6+7+y = 22+x+y$$
In order to be divisible by 9, that sum must either be 27 or 36.
The blanks must sum to $5$ or $14$.
It must also be divisible by $8$ so the last digit must be $2$. That means the first digit has to be $3$, because it would be impossible to make a sum of $14$.
Hence, the answer is x=3 and y=2
edited yesterday
PyThagoras
285
answered yesterday
Ross Millikan
279k23189355
You need the divisibility rules for $8,9$ because a number is divisible by $72$ only if it is divisible by $8$ and $9$ as well.
According to that the rule, the sum of the digits must be divisible by $9$.
$$x+6+7+y = 22+x+y$$
In order to be divisible by 9, that sum must either be 27 or 36.
The blanks must sum to $5$ or $14$.
It must also be divisible by $8$ so the last digit must be $2$. That means the first digit has to be $3$, because it would be impossible to make a sum of $14$.
Hence, the answer is x=3 and y=2
edited yesterday
PyThagoras
285
answered yesterday
Ross Millikan
279k23189355
edited yesterday
PyThagoras
285
edited yesterday
PyThagoras
285
edited yesterday
PyThagoras
285
285
answered yesterday
Ross Millikan
279k23189355
answered yesterday
Ross Millikan
279k23189355
answered yesterday
Ross Millikan
279k23189355
279k23189355
add a comment |Â
add a comment |Â
up vote
2
down vote
Without loss of generality, we can multiply the prices by 100, to work with integers and avoid the commas.
Let $n$ be the number of turkeys bought and $p$ be the price of each of them.
Then $$72p=10000a+679*10+b$$ where $a,b$ are between $0$ and $9$.
The trick is to notice that $72=9*8$. We can now use two divisibility tricks.
If a number is divisible by $8$, then the last 3 digits must be divisible by it (this follows by the fact that $1000$ is divisible by $8$). Hence $79*10+b$ must be divisible by $8$.
The only value of $b$ which satisfies this condition is $2$.
The second trick concerns divisibility by $9$: if a number is divisible by $9$, then the sum of the digits is divisible by $9$. Hence we must have $$a+6+7+9+2=24+a$$ to be divisible by 9.
The only value of $a$ which satisfies this condition is $3$.
To find the price, we calculate:
$$p=36792/72=511$$
Dividing back by 100, we get $p=5.11$ as you said.
answered yesterday
A-B-izi
12718
add a comment |Â
up vote
2
down vote
Without loss of generality, we can multiply the prices by 100, to work with integers and avoid the commas.
Let $n$ be the number of turkeys bought and $p$ be the price of each of them.
Then $$72p=10000a+679*10+b$$ where $a,b$ are between $0$ and $9$.
The trick is to notice that $72=9*8$. We can now use two divisibility tricks.
If a number is divisible by $8$, then the last 3 digits must be divisible by it (this follows by the fact that $1000$ is divisible by $8$). Hence $79*10+b$ must be divisible by $8$.
The only value of $b$ which satisfies this condition is $2$.
The second trick concerns divisibility by $9$: if a number is divisible by $9$, then the sum of the digits is divisible by $9$. Hence we must have $$a+6+7+9+2=24+a$$ to be divisible by 9.
The only value of $a$ which satisfies this condition is $3$.
To find the price, we calculate:
$$p=36792/72=511$$
Dividing back by 100, we get $p=5.11$ as you said.
answered yesterday
A-B-izi
12718
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Without loss of generality, we can multiply the prices by 100, to work with integers and avoid the commas.
Let $n$ be the number of turkeys bought and $p$ be the price of each of them.
Then $$72p=10000a+679*10+b$$ where $a,b$ are between $0$ and $9$.
The trick is to notice that $72=9*8$. We can now use two divisibility tricks.
If a number is divisible by $8$, then the last 3 digits must be divisible by it (this follows by the fact that $1000$ is divisible by $8$). Hence $79*10+b$ must be divisible by $8$.
The only value of $b$ which satisfies this condition is $2$.
The second trick concerns divisibility by $9$: if a number is divisible by $9$, then the sum of the digits is divisible by $9$. Hence we must have $$a+6+7+9+2=24+a$$ to be divisible by 9.
The only value of $a$ which satisfies this condition is $3$.
To find the price, we calculate:
$$p=36792/72=511$$
Dividing back by 100, we get $p=5.11$ as you said.
answered yesterday
A-B-izi
12718
Without loss of generality, we can multiply the prices by 100, to work with integers and avoid the commas.
Let $n$ be the number of turkeys bought and $p$ be the price of each of them.
Then $$72p=10000a+679*10+b$$ where $a,b$ are between $0$ and $9$.
The trick is to notice that $72=9*8$. We can now use two divisibility tricks.
If a number is divisible by $8$, then the last 3 digits must be divisible by it (this follows by the fact that $1000$ is divisible by $8$). Hence $79*10+b$ must be divisible by $8$.
The only value of $b$ which satisfies this condition is $2$.
The second trick concerns divisibility by $9$: if a number is divisible by $9$, then the sum of the digits is divisible by $9$. Hence we must have $$a+6+7+9+2=24+a$$ to be divisible by 9.
The only value of $a$ which satisfies this condition is $3$.
To find the price, we calculate:
$$p=36792/72=511$$
Dividing back by 100, we get $p=5.11$ as you said.
answered yesterday
A-B-izi
12718
edited 18 hours ago
answered yesterday
A-B-izi
12718
answered yesterday
A-B-izi
12718
answered yesterday
A-B-izi
12718
12718
add a comment |Â
add a comment |Â
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