# How many birds of each kind did I buy?

• Math100
In summary, by solving the Diophantine equation ## 10x+9y=180 ## and considering the constraints of the problem, it is determined that ten sparrows, nine turtle doves, and eleven doves were purchased, with each sparrow costing 3 pennies, each turtle dove costing 2 pennies, and each dove costing 2 pence. The relationship between pennies and pence is that one penny is equal to one hundredth of a pence.
Math100
Homework Statement
I bought sparrows at ## 3 ## for a penny, turtle doves at ## 2 ## for a penny, and doves at ## 2 ## pence each. If I spent ## 30 ## pence buying ## 30 ## birds and bought at least one of each kind of bird, how many birds of each kind did I buy?
Relevant Equations
Diophantine equation: ## ax+by=c ## where ## a, b, c ## are integers.
Let ## x ## denote the number of sparrows, ## y ## denote the number of turtle doves and ## z ## denote the number of doves.
Then we have ## \frac{1}{3}x+\frac{1}{2}y+2z=30 ## such that ## x+y+z=30 ##.
Observe that
\begin{align*}
&\frac{1}{3}x+\frac{1}{2}y+2(30-x-y)=30\\
&\frac{1}{3}x+\frac{1}{2}y+60-2x-2y=30\\
&-\frac{5}{3}x-\frac{3}{2}y=-30\\
&-10x-9y=-180\\
&10x+9y=180.\\
\end{align*}
Consider the Diophantine equation ## 10x+9y=180 ##.
By the Euclidean Algorithm, we have that ## gcd(10, 9)=1 ##.
Since ## 1\mid 180 ##, it follows that the Diophantine equation ## 10x+9y=180 ## can be solved.
From ## 1=10-1(9) ##, we get ## 180=180[10-1(9)]=180(10)-180(9) ##.
Thus ## x_{0}=-180, y_{0}=180 ##.
All solutions in the integers are determined by:
## x=-180+(\frac{10}{1})t=-180+10t ## for some ## t\in\mathbb{Z} ##,
## y=180-(\frac{9}{1})t=180-9t ## for some ## t\in\mathbb{Z} ##.
Thus, ## x=-180+10t ## and ## y=180-9t ##.
To find all solutions in the positive integers of the Diophantine equation ## 10x+9y=180 ##,
we solve the following inequalities for ## t ##:
## -180+10t\geq 0 ## and ## 180-9t\geq 0 ##.
Now we have ## 18\leq t\leq 20 ##, or ## t=19 ##.
This implies ## x=10, y=9 ##, and ## z=30-10-9=11 ##.
Therefore, ten sparrows, nine turtle doves and eleven doves were purchased.

Math100 said:
Homework Statement:: I bought sparrows at ## 3 ## for a penny, turtle doves at ## 2 ## for a penny, and doves at ## 2 ## pence each. If I spent ## 30 ## pence buying ## 30 ## birds and bought at least one of each kind of bird, how many birds of each kind did I buy?
Relevant Equations:: Diophantine equation: ## ax+by=c ## where ## a, b, c ## are integers.

Let ## x ## denote the number of sparrows, ## y ## denote the number of turtle doves and ## z ## denote the number of doves.
Then we have ## \frac{1}{3}x+\frac{1}{2}y+2z=30 ## such that ## x+y+z=30 ##.
Observe that
\begin{align*}
&\frac{1}{3}x+\frac{1}{2}y+2(30-x-y)=30\\
&\frac{1}{3}x+\frac{1}{2}y+60-2x-2y=30\\
&-\frac{5}{3}x-\frac{3}{2}y=-30\\
&-10x-9y=-180\\
&10x+9y=180.\\
\end{align*}
Consider the Diophantine equation ## 10x+9y=180 ##.
By the Euclidean Algorithm, we have that ## gcd(10, 9)=1 ##.
Since ## 1\mid 180 ##, it follows that the Diophantine equation ## 10x+9y=180 ## can be solved.
From ## 1=10-1(9) ##, we get ## 180=180[10-1(9)]=180(10)-180(9) ##.
This notation is new to me: ## 1=10-1(9) ## I have no idea what it means.
Math100 said:
Thus ## x_{0}=-180, y_{0}=180 ##.
All solutions in the integers are determined by:
## x=-180+(\frac{10}{1})t=-180+10t ## for some ## t\in\mathbb{Z} ##,
## y=180-(\frac{9}{1})t=180-9t ## for some ## t\in\mathbb{Z} ##.
Thus, ## x=-180+10t ## and ## y=180-9t ##.
To find all solutions in the positive integers of the Diophantine equation ## 10x+9y=180 ##,
we solve the following inequalities for ## t ##:
## -180+10t\geq 0 ## and ## 180-9t\geq 0 ##.
Now we have ## 18\leq t\leq 20 ##, or ## t=19 ##.
You should note that ##t\in \{18,20\}## is excluded by ##x,y>0.##
Math100 said:
This implies ## x=10, y=9 ##, and ## z=30-10-9=11 ##.
Therefore, ten sparrows, nine turtle doves and eleven doves were purchased.
Do you have a metal saw? Otherwise, you bought nine sparrows, ten turtle doves, and eleven doves.

Math100
fresh_42 said:
This notation is new to me: ## 1=10-1(9) ## I have no idea what it means.

You should note that ##t\in \{18,20\}## is excluded by ##x,y>0.##

Do you have a metal saw? Otherwise, you bought nine sparrows, ten turtle doves, and eleven doves.
I found my mistakes.

I'm not sure if I understand. You're using both pennies and pence. How do they relate to each other?

WWGD said:
I'm not sure if I understand. You're using both pennies and pence. How do they relate to each other?

'Penny' is either a currency unit worth 0.01 GBP, in which case the plural is 'pence', or a coin with a face value of one penny, in which case the plural is 'pennies'.

WWGD
@Math100, FWIW, you can simplify the working by utilising the fact that sparrows are bought in threes and turtle-doves are bought in twos.

S = number of sparrow triplets
T = number of turtle-dove doublets
D= number of doves

S+T+2D =30 (since total cost 30pence)
3S+2T+D =30 (since total no. of birds = 30)

A couple of lines of algebra gives:
5S+3T=30
which (in view of the small values) can easily be solved by inspection or by constructing a small table. (Though admittedly not a general/rigorous approach.)

## 1. How did you determine the number of birds of each kind that were bought?

The number of birds of each kind was determined through a scientific method, which involved collecting data, making observations, and analyzing the results.

## 2. What types of birds were bought?

The types of birds that were bought can vary, but some common examples could include parakeets, canaries, finches, lovebirds, and cockatiels.

## 3. How many total birds were bought?

The total number of birds bought depends on the specific scenario, but it can be calculated by adding up the individual numbers of each type of bird.

## 4. What was the purpose of buying the birds?

The purpose of buying the birds could vary, but some common reasons could include as pets, for breeding, or for scientific research.

## 5. How do you ensure the accuracy of your findings?

The accuracy of the findings is ensured through proper data collection and analysis techniques, as well as through repetition and peer review in the scientific community.

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