Help with finding X: A map scale indicates that 1/2 inch on the map corresponds with 3 real miles.

[EmmaR]

Hi everyone,

I'm an adult having to do math, after 20 years of not having had to do it.

Could someone please explain this equation? It makes zero sense to me, after staring at it for an hour and trying to understand.

 

[stapel]

Hi everyone,

I'm an adult having to do math, after 20 years of not having had to do it.

Could someone please explain this equation? It makes zero sense to me, after staring at it for an hour and trying to understand.

View attachment 37248

Has your class not yet covered ratios and proportions?

 

[Dr.Peterson]

Hi everyone,

I'm an adult having to do math, after 20 years of not having had to do it.

Could someone please explain this equation? It makes zero sense to me, after staring at it for an hour and trying to understand.

View attachment 37248

The equation is explained there, but clearly not in terms that you are prepared to understand.

In order to help better, we will need to know where you are struggling, and what you do understand, so we can start there (or at least recommend a source that can teach you the part(s) you need to know.)

Whatever you can tell about your context (such as whether you are taking a course, what that course is, and what you have learned recently), it will help us connect with you better. And the more you can tell us about your thinking as you try to work through the explanation, the better.

 

[Harry_the_cat]

A more logical way of thinking about it.

How many \(\displaystyle \frac{1}{2}\) inches are there in \(\displaystyle 2\frac{1}{4}\) inches? Can you see that there are \(\displaystyle 4\frac{1}{2}\)? (Look on a ruler if that helps.)

Now, each half inch represents 3 miles. So you have \(\displaystyle 4\frac{1}{2} \) lots of \(\displaystyle 3\) miles, ie \(\displaystyle 13\frac{1}{2}\) miles.

 

[Steven G]

2 1/4 = 1/2 + 1/2 + 1/2 + 1/2 + 1/4 (note that 1/4 is half of 1/2)
-----------3 + 3 + 3 + 3 + 1/2*3 (1/2 * 3 or half of 3 is 1.5
= 3 + 3 + 3 + 3 + 1.5 = 13.5

In my opinion you need to be able to do it this way before you can understand some of the methods above.

Also, you should try the same problems with friendlier numbers.

Suppose 2 inches refer to 5 miles. How many miles apart are two cities, if on the map they are 14 inches about.

For each 2 inches, the real distance is 5 miles. How many 2 inches are there in 14 inches? 14/2 = 7 gives you the answer to that question. Now you just need to multiply 7 by 5 miles to the the final answer of 35 miles.

 

[Harry_the_cat]

2 1/4 = 1/2 + 1/2 + 1/2 + 1/2 + 1/4 (note that 1/4 is half of 1/2)
-----------3 + 3 + 3 + 3 + 1/2*3 (1/2 * 3 or half of 3 is 1.5
= 3 + 3 + 3 + 3 + 1.5 = 10.5

In my opinion you need to be able to do it this way before you can understand some of the methods above.

Also, you should try the same problems with friendlier numbers.

Suppose 2 inches refer to 5 miles. How many miles apart are two cities, if on the map they are 14 inches about.

For each 2 inches, the real distance is 5 miles. How many 2 inches are there in 14 inches? 14/2 = 7 gives you the answer to that question. Now you just need to multiply 7 by 5 miles to the the final answer of 35 miles.

Adds to 13.5 not 10.5 (third line).
@ Steven G - Thought I'd mention so EmmaR is not confused.

 

[Steven G]

Adds to 13.5 not 10.5 (third line).
@ Steven G - Thought I'd mention so EmmaR is not confused.

It's been fixed. Thanks for pointing my error.

 

[The Highlander]

Hi everyone,

I'm an adult having to do math, after 20 years of not having had to do it.

Could someone please explain this equation? It makes zero sense to me, after staring at it for an hour and trying to understand.

View attachment 37248

When you say you want an explanation of the equation, it's not clear whether you are having difficulty in understanding how the equation was set up (created) or how it was manipulated to find a value for x (ie: how it was formulated or how it was solved).

Most of the above responses appear to assume your difficulty lies in grasping the concepts of Ratio & Proportion that underlie the construction of the equation and that can also be used to arrive at an answer (without algebraic manipulation of the equation).

I will, therefore, start by adding my own 'tuppence worth' in that area before addressing the equation itself.
Have a look at this table...

Alterations​	Distance on Map​	Represents this distance​	on real ground​
Halving​	¼​	1½​	95,040​
Given Ratio (as fraction)​	½ inch​	3 miles​	190,080 (inches)​
Doubling​	1​	6​	380,160​
Doubling again​	2​	12​	760,320​
Adding the 2" and the ¼"	2 + ¼ = 2¼​	12 + 1½ = 13½​	855,360​
(and just for info)​	\(\displaystyle \frac{1}{6}\)​	1​	63,360​

If you ignore the units (inches & miles; that's why I have only included units in the red row) you should see that the numbers in the the third column are all six times the numbers in the second column.
(Including 2¼ × 6 = 13½).

This means that the ratio of inches to miles is 1:6 (which can also be written as \(\displaystyle \frac{1}{6}\) ie: a fraction), so 1 inch represents 6 miles, 2 inches represent 12 miles and a ½ inch represents 3 miles (the ratio you were given).

The actual Scale of this map is 1:380 160. That means that 1 unit on the map represents 380,160 (of the same) units on the ground, so 1 inch represents 380,160 inches (ie: 6 miles) on the ground or 1 cm on the map would represent 380,160 cm (ie: c.3.8km) on the ground. Using that scale (1:380 160) means you can use any units: mm, cm, inches, or whatever.

Any ('good') map will report its Scale in this fashion but those kinds of numbers aren't much use to people so that's why they will include (or replace with on simpler maps) an 'explanation' like: "½ inch on the map corresponds with 3 real miles".

When I was in the army (many, many years ago) I was the squadron's appointed "Map Reading Officer", tasked with ensuring that all our men could read & use maps accurately. Most of our maps in those days used a Scale of 1:63 360 which was 1 inch to the mile (because a mile is 63,360 inches) but since decimalisation (in 1971) all our (modern) maps have switched to scales like 1:25 000. That means that, for example, 1 cm on the map represents 25,000 cm (250 m) on the ground or 4 cm on the map represents 1,000 m (ie: 1 km) or 1 inch represents 25,000 inches (c.0.4 miles/635 m).

In the UK, the Ordnance Survey (OS) is a governmental organization that is the national mapping agency responsible for producing accurate maps of the country and, in common with the rest of Europe, they now use metric scales as described above.

There is a very good explanation of all this on their website in the paragraph I have highlighted for you here that's definitely worth a read. (Just refresh the page if purple highlighting makes it difficult to read.)

A modern OS Map will have a Legend that show the Scale of the map (eg: 1:250 000) and (the more useful) 'equivalence' like: 1 cm to 2.5 km as shown below...

​

I explained above that the ratio of inches to miles (in:mi) is 1:6 and you can use that (simple) piece of information to arrive at a quick solution to the problem: if 1 in represents 6 mi then 2¼ in represent 2¼ × 6 = 13½ miles.

So that is basically a recap of what's been offered already (in an alternative fashion that I hope would make it clear if you still have any doubts). However, if it's the equation itself and how it was formed/solved that's causing your difficulties then it may be because they have used a couple of little shortcuts in their explanation that have, perhaps, bamboozled you?

Like the magician's prestidigitation, the hand can sometimes be quicker than the eye! So it may be useful to go over the algebraic manipulation in more detail to allow you to 'see' what is actually happening.

As I also explained above, that a ratio of 1:6 can also be written as the fraction \(\displaystyle \frac{1}{6}\). I trust you understand that if you take any fraction and multiply (or divide) the numerator (the top) and the denominator (the bottom) by the same number then you get an equivalent fraction.

Lets say we start with the fraction \(\displaystyle \frac{1}{6}\); you can see below that there are (infinite) equivalent fractions that all have exactly the same value...

\(\displaystyle \overbrace{\tiny{\qquad\qquad\qquad\qquad\qquad\qquad}}^{\overleftarrow{\text{Halving top \& bottom}}}\overbrace{\tiny{\qquad\qquad\qquad\qquad\qquad\qquad}}^{\overrightarrow{\text{Doubling top \& bottom}}}\overbrace{\tiny{\qquad\qquad\qquad\qquad\qquad\qquad}}^{\overrightarrow{\text{Doubling again}}}\\ \frac{\frac{1}{2}}{3}\qquad  =\qquad\qquad{\color{red}\frac{1}{6}}   \qquad= \quad\qquad\frac{2}{12}\qquad\quad = \qquad\frac{4}{24}\)​

Now it's very unusual to write a fraction where the numerator or the denominator (or both) is/are a fraction and so we would normally not write a result as \(\displaystyle \frac{\frac{1}{2}}{3}\) but would change that into the equivalent fraction of \(\displaystyle \frac{1}{6}\) but that is exactly what they have done to set up their equation; by saying that the ratio of \(\displaystyle \frac{1}{2}\) inch:3 miles is the same as: \(\displaystyle \frac{\frac{1}{2}\text{ map inch}}{\text{3 miles}}\qquad\left(\large{\text{ie: }}\small{\frac{\text{map inches}}{\text{real miles}}}\right)\)

Continued in the next post (there is a word limit to posts that I am about to exceed. )

 

[The Highlander]

Continued from previous post...

NB: You will need to log in to view this properly because I had to insert most of the algebraic manipulation as an image file (to avoid that pesky word limit! )

Now that they have established that the ratio (inches:miles) can be expressed as a fraction they can equate the 'new' map distance (2¼") to the corresponding real distance (they're calling "x") as an equivalent fraction which is how they came up with (formulated) the equation...

[math]\frac{\frac{1}{2}\text{ map inch}}{3\text{ miles}}=\frac{\text{2¼ map inches}}{x\text{ miles}}[/math]
So, let's now go through how they have manipulated that equation algebraically to solve for x...

The golden rule with equations is that you must do the same thing to both sides for it to remain true.

First, ignoring (removing) the units so we are just dealing with numbers, you get...

[math]\frac{\frac{1}{2}}{3}=\frac{\text{2¼}}{x}[/math]

​

Please come back and tell us if you have any further difficulties or do you now understand what's been going on?

Hope that helps.