# Problem Solving with Linear Equations

#### Jonnydog

##### New member
I need help with the following questions:

1. One half my age is 10 years more than one-third my age. How old am I?

2. A maker of an orange drink can purchase her raw materials from two sources.
The first source provides liquid with 6% orange juice, while the second source provides liquid with 3% orange juice. She wishes to make 1 litre of drink with 5% orange juice. Lex x = amount of liquid purchased from the first source.

A. Write the expression for the amount of orange juice from the first supplier, given that x is the amount of liquid

B. Write an expression for the amount of liquid from the second supplier, given that x is the amount of liquid used from the first supplier.

C. Write an expression for the amount of orange juice from the second supplier.

D. Write an equation for the total amount of orange juice of the mixture of the 2 supplies, given that 1 litre of drink is mixed to contain 5% orange juice.

E. How much of the first supplier's liquid should she use?

3. Rachel, the bushwalker, goes on a 4-day journey. She travels a certain distance on the first da, half that distance on the second day, a third that distance on the third day and a fourth of that distance on the fourth day. If the total journey is 50km, how far did she walk on the first day?

Thanks,
Jonnydog

#### jonboy

##### Full Member
Hello, Jonnydog!

Jonnydog said:
I need help with the following questions:

1. One half my age is 10 years more than one-third my age. How old am I?
First step it to name stuff. We will call $$\displaystyle a$$ your age.

One half my age is 10 years more than one-third my age:$$\displaystyle \L \;\frac{1}{2}a\,=\,\frac{1}{3}a\,+\,10$$

Subtract $$\displaystyle \frac{1}{3}a$$:$$\displaystyle \L \;\frac{1}{6}a\,=\,10$$

Divide by $$\displaystyle \frac{1}{6}$$:$$\displaystyle \L \;a\,=\,60$$

Check:$$\displaystyle \L \;(\frac{1}{2})(60)\,=\,(\frac{1}{3})(60)\,+\,10\,\to\,30\,=\,30$$

Alright there is one for you. Just name stuff, set up the equations, and then solve the system by the appropriate method.

¡Chao!

#### Jonnydog

##### New member
Thanks, I understand that now, but I've still made no progress on the other ones. =(

#### jonboy

##### Full Member
Jonnydog said:
Thanks, I understand that now, but I've still made no progress on the other ones. =(

#### Jonnydog

##### New member
With 2, I have no idea whatsoever.

With 3, here's what I've done

a + 1/2 * a + 1/3 * a + 1/4 * a = 50

then I just kind of put them together
a + 13/12 * a = 50

and I don't know if it's right or not, and if it is, I don't know where to go from here

#### pka

##### Elite Member
This is not a homework service!
We do not do your homework for you!
We do not do teach you the material.
So show us what you have tried.

#### Jonnydog

##### New member
....I just posted what I had tried.

#### jonboy

##### Full Member
Sry for number 3 you are wrong.

3. Rachel, the bushwalker, goes on a 4-day journey. She travels a certain distance on the first da, half that distance on the second day, a third that distance on the third day and a fourth of that distance on the fourth day. If the total journey is 50km, how far did she walk on the first day?
Let $$\displaystyle a$$ equal the distance.

So:$$\displaystyle \L \;a\,+\,\frac{1}{2}a\,+\,\frac{1}{3}a\,+\,\frac{1}{4}a\,=\,50$$

Simplify:$$\displaystyle \L \;2\frac{1}{12}a\,=\,50$$

Divide by $$\displaystyle 2\frac{1}{12}$$:$$\displaystyle \L \;a\,=\,24$$

So the distance of the first day is 24 kilometer. Does that make since?

#### jonboy

##### Full Member
The 4 day is irrelevant because you don't know how much of each day Rachel travels. It just says a distance.

#### Jonnydog

##### New member
Okay I think I get it, so you divide 50 by 25/12?

#### jonboy

##### Full Member
Jonnydog said:
Okay I think I get it, so you divide 50 by 25/12?
Yes. Since all of those distances add up to 50, I just added them altogether and set it equal to 50 km. Then solved for a, and a is just the first distance.

#### Jonnydog

##### New member
Okay, thanks for the help with #3, I understand now =)

With number 2, I am almost completely clueless, except for A.

A. Juice = 6/600 * x, so 0.06x right?

#### Denis

##### Senior Member
Jdog, you can make #3 real easy by letting 1st day = 12a;
then 2nd = 6a, 3rd = 4a and 4th = 3a; so you have:
12a + 6a + 4a + 3a = 50
25a = 50
a = 50/25 = 2

So 1st day = 12a = 12 * 2 = 24 ; capish?

#### Denis

##### Senior Member
Jonnydog said:
A maker of an orange drink can purchase her raw materials from two sources.
The first source provides liquid with 6% orange juice, while the second source provides liquid with 3% orange juice. She wishes to make 1 litre of drink with 5% orange juice. Lex x = amount of liquid purchased from the first source.
A. Write the expression for the amount of orange juice from the first supplier, given that x is the amount of liquid
B. Write an expression for the amount of liquid from the second supplier, given that x is the amount of liquid used from the first supplier.
C. Write an expression for the amount of orange juice from the second supplier.
D. Write an equation for the total amount of orange juice of the mixture of the 2 supplies, given that 1 litre of drink is mixed to contain 5% orange juice.
E. How much of the first supplier's liquid should she use?
RULE; IF:
x @ a
y @ b
=======
(x+y) @ c
THEN:
(ax + by) / (x + y) = c

x @ 6
(1-x) @ 3
=======
1 @ 5

(6x + 3(1-x)) / 1 = 5
6x + 3 - 3x = 5

Can you finish?