# Direct Proportion, algebraic formula

#### bumblebee123

##### Junior Member
Hello,

Please, could someone help me out with a question in my daughters' Math book. The question relates to direct proportion and is given in the form of information. It seems quite straightforward and I see how they get the answer but it's not how I get the answer and I don't know why they choose to put the letters where they do. The answer I get is wrong but why? I have copied it out exactly:

Dave is working out different ways of travelling around Europe. He is making maps showing the cities he might visit. The time, T minutes, that it takes Dave to draw a map is directly proportional to the square of the number of cities, c, he puts on a map. A map with 8 cities takes 10 minutes to draw.

Question a) Find a formula for T in terms of c.

T = kc^2

( I use ^2 to stand for the square, so its c squared)

I substitute the known numbers for cities and time taken to draw:
10 = k8^2

I continue to solve:

10 = k64

10/64 = k

Substitute the k in to the formulae to give the answer:

T = 10/64c^2

so, T is equal to 10 divided by 64 multiplied by c squared.

T = c^2/6.4

So I figure they have done: c^2 = kT

in order to get their answer. But why have they done this when the question says to find a formula for T in terms of c? How would I know to put the constant of proportionality next to Time and not cities?

Your thoughts would be much appreciated.

#### mmm4444bot

##### Super Moderator
Staff member
… T = 10/64 c^2

T = c^2/6.4
These two expressions for T are different forms of the same answer. In other words, dividing by 6.4 is the same as multiplying by 5/32 (note that I reduced your fraction to lowest terms).

The rules are: (1) Dividing by a fraction is the same as multiplying by its reciprocal. (2) Multiplying by a fraction is the same as dividing by its reciprocal.

Rewrite 6.4 as an improper fraction, and you'll see it! :cool:

#### stapel

##### Super Moderator
Staff member
Please, could someone help me out with a question in my daughters' Math book.
Unfortunately, many of us have learned from hard experience that attempting to tutor through a "translator" who doesn't "speak the language" has very little chance of success. It would be much better if we could speak with the student directly.

The answer I get is wrong but why? I have copied it out exactly:

Dave is working out different ways of travelling around Europe. He is making maps showing the cities he might visit. The time, T minutes, that it takes Dave to draw a map is directly proportional to the square of the number of cities, c, he puts on a map. A map with 8 cities takes 10 minutes to draw.

Question a) Find a formula for T in terms of c.

T = kc^2

I substitute the known numbers for cities and time taken to draw:
10 = k8^2

I continue to solve:

10 = k64

10/64 = k

Substitute the k in to the formulae to give the answer:

T = 10/64c^2
As written, this would be interpreted as:

. . . . .$$\displaystyle T\, =\, \dfrac{10}{64c^2}$$

However, I suspect that you mean:

. . . . .$$\displaystyle T\, =\, \dfrac{10}{64}\, c^2$$

This can be simplified as:

. . . . .$$\displaystyle T\, =\, \dfrac{5}{32}\, c^2$$

so, T is equal to 10 divided by 64 multiplied by c squared.

T = c^2/6.4
Usually, books (and instructors) prefer the fractional form, but there is nothing wrong with converting "10/64" to "1/6.4"; just divide top and bottom by 10.

#### bumblebee123

##### Junior Member
Thank you both for the help, much clearer now.
Wish I knew how to write it like that on the computer, I must be missing a button with symbols on.

#### mmm4444bot

##### Super Moderator
Staff member
Wish I knew how to [post mathematical formatting] like that [in the forum.]

I must be missing a button with symbols on.
No, there's no button. It's a typesetting system known as LaTex. Its use requires learning programming codes and syntax. You can google for tutorials and examples, but, unless you plan on using it a lot at forums, it may not be worth the effort for you.

In the forum guidelines, there's a link to a page that shows how to format math as text. That's an easy alternative to learning LaTex.

If you'd like to view the coding for any LaTex you see rendered here, you may right-click on a LaTex expression and use the pop-up menu to Show Math As → TeX Commands.

At this site, each line of LaTex coding must be enclosed within $$\displaystyle \text{[te}\text{x]}$$ and $$\displaystyle \text{[/te}\text{x]}$$ tags. :cool: