linear equation

[Rose16]

Kara starts tying a baby quilt at the local community center. Each quilt takes 200 ties to complete. Ten minutes later Julie joins Kara, and both stay to finish the quilt. Kara ties 6 per minute and Julie can tie 11 per minute.

a. Write an expression to show the number of ties Kara has completed after m minutes.
b. Using the same variable m, write an expression for the number of ties Julie has completed.
c. If you know they finished the quilt, write an equation to model the situation.
d. How many minutes did it take them to finish the quilt?
e. How many did each of them tie?
f. Suppose they tied the same amount. How long would it take?
g. How many did each of them tie?

h. The next week, Julie couldn’t make it so her friend Susan came to help. Susan can tie the ties ¾ as fast as Kara. Using the same variable m, write an expression for the number of ties Susan has completed.
1h. If you know they finished a quilt, write an equation to model the situation.
2h. How many minutes did it take them to finish the quilt?
3h. How many did each of them tie?
solution;

This is what I have so far. I am stuck because I don't know if I should subtract the 60 ties Kara did before writing my equation.

 

[Deleted member 4993]

Kara starts tying a baby quilt at the local community center. Each quilt takes 200 ties to complete. Ten minutes later Julie joins Kara, and both stay to finish the quilt. Kara ties 6 per minute and Julie can tie 11 per minute.

a. Write an expression to show the number of ties Kara has completed after m minutes.
b. Using the same variable m, write an expression for the number of ties Julie has completed.
c. If you know they finished the quilt, write an equation to model the situation.
d. How many minutes did it take them to finish the quilt?
e. How many did each of them tie?
f. Suppose they tied the same amount. How long would it take?
g. How many did each of them tie?

h. The next week, Julie couldn’t make it so her friend Susan came to help. Susan can tie the ties ¾ as fast as Kara. Using the same variable m, write an expression for the number of ties Susan has completed.
1h. If you know they finished a quilt, write an equation to model the situation.
2h. How many minutes did it take them to finish the quilt?
3h. How many did each of them tie?
solution;
View attachment 33685
This is what I have so far. I am stuck because I don't know if I should subtract the 60 ties Kara did before writing my equation.

Before you start to solve this set of equations, define your variables.

What is k?......What is j?.......... What is t? (m is already defined in the problem)

"Susan can tie the ties ¾ as fast as Kara." - Does this mean Susan can tie 3 knots in the same time that Kara tied 4 knots (implying Susan is slower) or something else?

 

[JeffM]

Rose

Please think about your answer to a.

How many ties can Kara do in m minutes if she can do 6 in one minute?

 

[Steven G]

It is not possible to tell if part c is correct because you have not defined your variables. You especially have to define t carefully. Also, part a and p are not correct. Plug in various haves for t and you'll see what is wrong.

 

[Rose16]

Rose

Please think about your answer to a.

How many ties can Kara do in m minutes if she can do 6 in one minute?

If 1 minute =6 ties
m=m/1 *6
.:she can do 6m

 

[JeffM]

If 1 minute =6 ties
m=m/1 *6
.:she can do 6m

Exactly. So do you want to rethink your answers to questions a, b, and c?

And please say whether t represents number of ties or time required?

 

[Rose16]

Exactly. So do you want to rethink your answers to questions a, b, and c?

And please say whether t represents number of ties or time required?

 

[Steven G]

Let's look at k(t) = 6m. Note that there no t's on the right hand side of the equal sign. That means that the right hand side does not change as t changes.

For example, if t=5, then k(5) = 6m. If t=200, then k(200) = 6m

How about if you used k(m) = 6m, will that work??

 

[Steven G]

m=1, 6ties
m=2, 6*2ties = 12 ties
m=3, 6*3ties

So k(m) = (???)ties

To find out how ties k has done, how do YOU figure it out? Let m equal your answer.

 

[Otis]

Let t represent the number of ties
Let k represent Kara
Let j represent Julie

Hi Rose. That's a good definition for symbol t. It shows that t represents a count of something specific.

Symbols k and j also represent numbers, but the definitions above don't specify what. We can understand what you meant, but it would be better form to state the meaning in the definitions – instead of defining k and j as human females.

And, thank you for showing your work. (I wish more students would.)


[imath]\;[/imath]

 

[Rose16]

My question is why his solution is different from mine. Did I do something wrong. I am not sure if I am right or not

 

[Steven G]

Lets look at a:
t = # of ties
Suppose Kara works for 5 minutes. She completes 6ties per minute. In the end, she will have made 30 ties. Is it clear up to here?
Now you say, K(t) = \(\displaystyle 6\dfrac{ties}{minute}*t\)

Now you are being asked how many ties can Kara make in m minutes, ie you are being asked to find t. To compute K(t), you need to know t. But we don't know t. We are trying to find t. So your formula must be wrong. You must understand what I wrote above.

To find t, your formula must start off with t =. The right hand side can't have t's in it.

Also check you units. Since you want to know how many ties there are, then the right hand side should just have the unit ties.

In your formula \(\displaystyle 6\dfrac{ties}{minute}*t\) the units will be \(\displaystyle \dfrac{ties}{minute}*ties = \dfrac{ties^2}{minute}\), since the units of t is ties

 

[Steven G]

You know that Kara makes 6 ties every minute. To figure out how many ties Kara made altogether what do you need to know??????

 

[Rose16]

You know that Kara makes 6 ties every minute. To figure out how many ties Kara made altogether what do you need to know??????

The total number of ties used to make the quilt. And the number of ties doe by Julie?

 

[Rose16]

Lets look at a:
t = # of ties
Suppose Kara works for 5 minutes. She completes 6ties per minute. In the end, she will have made 30 ties. Is it clear up to here?
Now you say, K(t) = \(\displaystyle 6\dfrac{ties}{minute}*t\)

Now you are being asked how many ties can Kara make in m minutes, ie you are being asked to find t. To compute K(t), you need to know t. But we don't know t. We are trying to find t. So your formula must be wrong. You must understand what I wrote above.

To find t, your formula must start off with t =. The right hand side can't have t's in it.

Also check you units. Since you want to know how many ties there are, then the right hand side should just have the unit ties.

In your formula \(\displaystyle 6\dfrac{ties}{minute}*t\) the units will be \(\displaystyle \dfrac{ties}{minute}*ties = \dfrac{ties^2}{minute}\), since the units of t is ties

could my formula be k(t)=6m

 

[JeffM]

I must say that I found the way this problem and answer have been presented to be very confusing for a beginning student. Why bring function notation into the picture? The unfortunate student has clearly not even been taught to start by naming relevant quantities. And why in the world give a problem where any answer that is not in integers makes no sense?

[math] k = \text {time spent by Kara in minutes;}\\ j = \text {time spent by Julie in minutes;}\\ p = \text {ties completed by Kara;}\\ q = \text {ties completed by Julie; and}\\ t = \text {time spent doing the quilt.}[/math]
We are told that: Kara started 10 minutes before Julie, Kara can do 6 ties per minute, Julie can do 11 ties per minute, and that a quilt requires 200 ties.

The whole discussion about m is confusing. Clearly, in m minutes, Kara can do 6m ties whereas Julie can do 11m ties. So what? The first part of the problem specifies that they do not spend the same amount of time.

We have five unknowns so we need five equations.

p + q = 200. Why?

k = 10 + j. Why?

p = 6k. Why?

q = 11j. Why?

t = k. Why?

Now everything is purely mechanical.

[math] p = 6(10 + j) = 60 + 6j.\\ q = 11j.\\ \text {Thus, } 200 = p + q = 60 + 6j + 11j = 60 +17j \implies \\ 17j = 200 - 60 = 140 \implies j = \dfrac{140}{17} \approx 8.24. \\ \text {And } k = 10 + j \implies k = 10 + \dfrac{140}{17} = \dfrac{170 + 140}{17} = \dfrac{310}{17} \approx 18.24.\\ t = k \approx 18.24.\\ p = 6 * \dfrac{310}{17} \approx 109.4.\\ q = 11 * \dfrac{140}{17} \approx 90.6.\\ [/math]