Patterns, Relationships, & Algebraic Thinking - Need Help?

Question

I'm trying to help My son out with his homework and I can't.. Sad to say... He has 20 problems and if someone can help explain one to me I should be able to help him with the reset.. I just need a guideline to go by..

Question 1 -

A map key shows that every 3 centimeters represents 60 miles. Which proportion could be used to determine the number of miles represented by 8 centimeters?

A - 60/3 = 8/m
B - 3/8 = m/60
C - 3/5 = 60/m
D - 3/60 = 8/m

I need to know HOW TO GET THE ANSWER not just the answer.. This way I can help him on the rest...

Thanks a bunch to anyone that can help..

Answer 1

The relationship between centimeters and miles can be expressed as a ratio, written 3/60 or 3:60. This simply means that for every 3 centimeters, there are 60 miles represented. Once we've established that ratio, we can manipulate it.

For instance, I can say that 1 centimeter represents 20 miles, because (3/3 = 1) and (60/3 = 20). I could also say that 300 centimeters represent 6000 miles, because (3 * 100 = 300) and (60 * 100 = 6000). As long as I perform the same operations on both sides of the ratio, the equation will remain true.

The problem asks what proportion could be used to determine distance represented by eight centimeters. Basically, they are asking you to express the ratio in a different manner. If we used the variable "m" to represent the unknown quantity of miles, the problem would look like this:

3 centimeters / 60 miles = 8 centimeters / m miles. If we eliminate the units, in order to simplify, we get 3/60 = 8 / m. Thus "D" is the correct answer to the question.


In order to solve a problem of this sort, we have to figure out what we have to multiply 3 by to get 8. Once we know that, we can perform the same function on 60 to get "m".

We can figure out that number by making a formula: (3 * x = 8). Because division and multiplication are inverse fuctions, we can work the formula backwards to solve for x: (x = 8/3). So 8/3 is the number that we multiply by to turn 3 into 8. If we multiply 60 by the same number, it will turn into "m". (60 * 8/3 = 480/3 = 160), so m = 160 miles.

The quick way to do this is by cross-multiplying. If I have a formula of the construction (3/60 = 8/m), I can solve it by multiplying the two known values that are diagonal from one another (60 and 8), then dividing that number by the remaining known value (3). The resulting number will be the unkown value. The equation looks like this (8*60/3=m). Learning to set up these ratio formulas and cross-multiply will allow you to solve ratio problems, like this one, much more quickly.

Hope this helped. Good luck.

Answer 2

You need to first figure out what you are comparing. For example: centimeters / miles (let / = over)

Then you want to set up a proportion. If you know that 3 cm / 60 miles (centimeters being on the top and miles being on the bottom, you need to remember to be consistent) then 8 centimeters will go on the top.

Now you have 3/60 = 8/x (or any variable which is a letter that represents a #).

So the answer would be D.

Hope this helps.

Answer 3

D - 3/60 = 8/m

cross multiply

60*8 = 3*m

Solve for m

m = 60*8/3
m = 160 miles

Answer 4

This is a matter of ratio.

if 3 cm is to 60 miles, then 8 cm is to x

so...x/8 = 60/3

x= 60/3x8

x = 160 miles!

m=x

D!!!!!

Answer 5

the answer is A.