# Equations word problems (1 Viewer)

##### Member
Please can you show working for the following:

a) At a sale, Sarah spent $143 on some shirts and shorts. The shirts cost$15 each and the shorts cost $17 each. What is the total number of shirts and shorts that Sarah bought? b) Bill sits for a test containing 20 questions. He gets 2 marks for each correct answer and loses 1 mark for each incorrect answer. He scored 22 marks in the test. How many questions did he get correct? c) Fred is twice as old as his son and Fred is 36 years older than his son. How old is Fred? d) Mina earns$3600 more than Betty and Lina earns $2000 less than Betty. If the total of the three incomes is$151 600, find the incomes of each person.

#### Etho_x

##### Well-Known Member
a didn’t stick out to me at first so I did b, c, and d first lol

#### Etho_x

##### Well-Known Member
For a though, I thought it was just easier to do process by elimination since it would be a reasonably small number for both. We know 15A + 17B = 143, where A is the number of shirts and B is the number of shorts bought.

Rearranging for A: (143-17B)/15 = A
By process of elimination, the result yielding both whole numbers is when B = 4 and A = 5, that is, the person bought 5 shirts and 4 shorts.

##### Member
For a though, I thought it was just easier to do process by elimination since it would be a reasonably small number for both. We know 15A + 17B = 143, where A is the number of shirts and B is the number of shorts bought.

Rearranging for A: (143-17B)/15 = A
By process of elimination, the result yielding both whole numbers is when B = 4 and A = 5, that is, the person bought 5 shirts and 4 shorts.
What will the second equation be?

#### Etho_x

##### Well-Known Member
What will the second equation be?
Well that’s what I was thinking, I feel as if there’s information missing but then again I’m not sure. That’s how I did it

#### Qeru

##### Well-Known Member
Please can you show working for the following:

a) At a sale, Sarah spent $143 on some shirts and shorts. The shirts cost$15 each and the shorts cost $17 each. What is the total number of shirts and shorts that Sarah bought? b) Bill sits for a test containing 20 questions. He gets 2 marks for each correct answer and loses 1 mark for each incorrect answer. He scored 22 marks in the test. How many questions did he get correct? c) Fred is twice as old as his son and Fred is 36 years older than his son. How old is Fred? d) Mina earns$3600 more than Betty and Lina earns $2000 less than Betty. If the total of the three incomes is$151 600, find the incomes of each person.
$\bg_white 15x+17y=143$ where x and y are positive integers. This is a linear diophantine equation. For the purposes of HSC guess and check works. However the way we solve these types of equations is as follows:
Take the euclidean algorithm to find gcd(15,17):
17 = 15(1) + 2
15 = 2(7) + 1
2 = 1(2) + 0
So gcd(15,17)=1
Then write 1 as a linear combination of 15 and 17 (by reversring the euclidean algorithm above):
i.e. 1=15-2(7)=15-(17-15)(7)=8(15)-7(17)

So: $\bg_white 1=8(15)-7(17)$

Then simply multiply both sides of the equation by 143 to get:

$\bg_white 143=1144(15)-1001(17)$
So a solution is: $\bg_white x=1144,y=-1001$
But notice how: $\bg_white 143=15(1144-17n)+17(-1001+15n)$
So our general sols are: $\bg_white x=1144-17n, y=-1001+15n$ where n is any integer.
Now we want positive x and y solutions so: $\bg_white 1144-17n>0 \implies n\leq67$ and $\bg_white -1001+15n>0 \implies n \geq 67$. So our only solution is when n=67 i.e. $\bg_white (1144-17(67),-1001+15(67))=(5,4)$.

#### Etho_x

##### Well-Known Member
$\bg_white 15x+17y=143$ where x and y are positive integers. This is a linear diophantine equation. For the purposes of HSC guess and check works. However the way we solve these types of equations is as follows:
Take the euclidean algorithm to find gcd(15,17):
17 = 15(1) + 2
15 = 2(7) + 1
2 = 1(2) + 0
So gcd(15,17)=1
Then write 1 as a linear combination of 15 and 17 (by reversring the euclidean algorithm above):
i.e. 1=15-2(7)=15-(17-15)(7)=8(15)-7(17)

So: $\bg_white 1=8(15)-7(17)$

Then simply multiply both sides of the equation by 143 to get:

$\bg_white 143=1144(15)-1001(17)$
So a solution is: $\bg_white x=1144,y=-1001$
But notice how: $\bg_white 143=15(1144-17n)+17(-1001+15n)$
So our general sols are: $\bg_white x=1144-17n, y=-1001+15n$ where n is any integer.
Now we want positive x and y solutions so: $\bg_white 1144-17n>0 \implies n\leq67$ and $\bg_white -1001+15n>0 \implies n \geq 67$. So our only solution is when n=67 i.e. $\bg_white (1144-17(67),-1001+15(67))=(5,4)$.