
Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion 
 LinkBack  Thread Tools  Display Modes 
August 10th, 2014, 08:56 PM  #1 
Senior Member Joined: Feb 2014 From: Louisiana Posts: 156 Thanks: 6 Math Focus: algebra and the calculus  Division for ratios
Why is division used for ratios? For example, if we have 1 dog for every 2 cats, then why can this relation be modeled by the division? Since the ratio can essentially have the value of 1/2 or 2, then what is the significance of these two distinct numbers in relation to the ratio?

August 11th, 2014, 05:22 AM  #2 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,155 Thanks: 731 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
Ratios are about proportions, which are related to fractions. If you have 1 dog for every 2 cats, then that means that 1 animal out of 3 is a dog and 2 animals out of 3 are cats. In other words, $\displaystyle \frac{1}{3}$ of all animals are dogs and $\displaystyle \frac{2}{3}$ of all animals are cats. So... the division aspect comes in because fractions are equivalent to unsolved division problems: $\displaystyle \frac{1}{3} = 1 \div 3$ $\displaystyle \frac{2}{3} = 2 \div 3$ Note: only cats or dogs are included in my definition of animal here! Also, you might be wondering about the representation of ratios in the form $\displaystyle 1:2 = \frac{1}{2}:1$ These are equivalent ratios and are exactly the same. The ratios will remain equivalent provided you multiply or divide both sides by the same number, but for the sake of neat presentation it is convenient to use 'nice' representations such as a:b where a and b are integers, or 1:n where n is a real number. 
August 11th, 2014, 05:39 AM  #3  
Senior Member Joined: Feb 2014 From: Louisiana Posts: 156 Thanks: 6 Math Focus: algebra and the calculus  Quote:
 
August 11th, 2014, 07:36 AM  #4 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,155 Thanks: 731 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
Each individual number on its own in a ratio is meaningless; you must have at least two numbers so you can compare them. Then all you care about is the relative size of the two or more numbers. Order of numbers matters. If you wanted to communicate to a mathematician the particular ratio you gave "1 dog to 2 cats", I could use any of the following choices: 1:2 2:4 3:6 50:100 100:200 0.5:1 0.01:0.02 It really doesn't matter. All of these give the exact same proportion and are called equivalent ratios. If you swap the order, you need to remember to swap the thing it describes too, so it reads "2 cats to 1 dog", instead of "2 dogs to 1 cat" which is obviously a different ratio. 
August 11th, 2014, 08:09 AM  #5  
Senior Member Joined: Feb 2014 From: Louisiana Posts: 156 Thanks: 6 Math Focus: algebra and the calculus  Quote:
 
August 11th, 2014, 08:22 AM  #6 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  
August 12th, 2014, 01:24 AM  #7  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,155 Thanks: 731 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
It doesn't. You have two choices to represent the specific ratio you specified: 1) Use a colon So.... ratio of dogs to cats is 1:2 (or any of its equivalents) 2) Find the fraction of each to the total $\displaystyle \frac{1}{3}$ are dogs and $\displaystyle \frac{2}{3}$ are cats. That's it.  
August 12th, 2014, 08:14 AM  #8 
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 390 Thanks: 70 
$\displaystyle \frac {1}{2}$ is an alternate symbol for the ratio 1:2, and it is not a fraction when it is used in that way. One reason that its a good alternate symbol is that when the symbol is considered to represent a fraction, it has the same equivalences as when used to represent a ratio. For example: $\displaystyle \frac {1}{2}$=$\displaystyle \frac {2}{4}$=$\displaystyle \frac {6}{12}$=$\displaystyle \frac {18}{36}$ These equivalences hold both as fractions and as ratios. Using the standard ratio symbols: 1:2 = 2:4 = 6:12 = 18:36 
August 12th, 2014, 02:01 PM  #9 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 2,001 Thanks: 132 Math Focus: Trigonometry 
I remember when teaching some $\displaystyle 7^{st}$ grader students, there was a question like this in their books: What is the difference between $\displaystyle \frac{1}{2}$, 1/2, and 1:2? 
August 13th, 2014, 02:17 AM  #10  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,155 Thanks: 731 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
Quote:
For example, consider the following two common exam questions, most of which use the knowledge of equivalent fractions without having to introduce the notation above. I will outline explicitly three possible methods of solution of the first question, all of which have no need to introduce the 1/2 notation. Question 1. "Dogs and cats are found in the ratio 1:2. If 15 animals are found, how many cats are there?". First step (all methods) 1+2 = 3 Therefore, $\displaystyle \frac{2}{3}$ of all animals are cats. Method 1 $\displaystyle 15 \div 3 = 5$, so there are 5 times as many animals than 3. $\displaystyle \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$ Therefore, 10 cats are found. Method 2 (my preferred method) Find $\displaystyle \frac{2}{3}$ of 15 $\displaystyle \frac{2}{3}$ of 15 $\displaystyle \frac{2}{3} \times 15$ $\displaystyle 15\div 3 = 5$ $\displaystyle 5 \times 2 = 10$ Therefore 10 cats are found. Method 3 $\displaystyle 15 \div 3 = 5$, so there are 5 times as many animals than 3. 1:2 $\displaystyle 1 \times 5 : 2 \times 5$ 5:10 Therefore, 10 cats are found. Question 2. "A recipe for "Butterfly bread" contains 200g of flour, 50g of butter and 20g of sugar, which serves one person. A cook makes a some Butterfly bread that contains 1080g of these three ingredients. How many people can be served?" 200:50:20 200 + 50 + 20 = 270 $\displaystyle \frac{\cancel{1080}^{108}}{^{27}\cancel{270}} = \frac{\cancel{108}^{4}}{^{1}\cancel{27}} = 4$ So four people can be served. Usually a student is asked to focus on one of these methods and stick with it, but it's not unusual for a student to be familiar enough with fractions such that they figure out the various ways they can manipulate them by themselves. Method 3 is useful if a student needs to know the number of dogs too. Please discard this 1/2 thing from your mind. I'm tired of seeing people put 1/2 for questions like this and then wonder why they get the answer wrong when they put 1/2 of 15 = 7.5 cats. In addition, ratios can have more than 2 numbers, as in question 2 above. Putting 200/50/20 is unnecessary and confusing.  

Tags 
division, ratios 
Search tags for this page 
math problems ratios of cats to dogs,the ratio of dogs to cats in simons neighboorhood is 4:6. if their are 36 cats how many dogs?,division for ratio,division ratios,division of a quantity in a particular ratio problems
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
ratios here  dante  Elementary Math  2  May 14th, 2014 07:41 AM 
If revenue's at the Zurich division were less than those at Prague Division, then Zur  pnf123  Advanced Statistics  0  March 29th, 2014 05:26 AM 
Ratios  linhbui  Elementary Math  4  August 22nd, 2012 06:24 AM 
Help with Trignometric Ratios  jattrockz  Trigonometry  5  November 10th, 2011 12:17 PM 
Ratios  MathematicallyObtuse  Algebra  3  January 9th, 2011 08:07 AM 