Ratios

Ratio

Introduction

A ratio is nothing but a comparison between two numbers by division.
Generally, we know comparing as to tell if a number is <, > or =, but we are not trying to compare number like that, instead, we are really trying to see how two numbers relate to each other.

ratios usage in real-life:

So to know how we show the relationship between two numbers by division here is an example…
take the ratio 6/9, but remember that we can write a ratio in the form for a fraction.

you can also write a ratio using the ‘: ‘ ex. 6:9 which means  6 to 9 or 6 per 9 or 6 is to 9.

(I recommend saying per because it is clear & easy to understand)

The difference between fractions and ratios

fractions and ratios are basically the same things, it is just that when we use the fraction in a particular way it is called a ratio.  A fraction is a single number.

Example: have a sandwich for lunch but that day you were not so hungry so you eat half of a sandwich. Here half of a sandwich is a fraction of a whole sandwich.

In the case of a ratio, we have to concentrate on both top and bottom numbers because a ratio is not one number.

Example: let us say that two people went to a picnic and they bought one sandwich per two people, and one sandwich per two people is a ratio. So the ratio is 1:2

In our example 1 and 2 are referring to two different things, 1 is referring to the sandwich and 2 refers two the people.

Simplifying a Ratio

why would we want to simplify a ratio

Simplifying ratios give a clear understanding of the relationship between the two things or number.
Simplifying ratios make them easier to work with.

Find the simplest form of 30:40

First, find out what is the common multiples of 30 and 40.

2,5,10 are common multiples for 30,40  3 is a common multiple.

so then divide the 30 and 40 with common multiples.

30÷2=15
30÷5=6
30÷10=3
40÷2=20
40÷5=8
40÷10=4

so the equivalent ratio is 15:20,6:8,3:4 so the simplest ratio from of 30:40 is 3:4.

To get the directly the smallest ratio, divide the actual ratio with the greatest common multiple.

Proportions

proportions are equivalent Ratios.

To know what proportions are here is an example…

Let us assume that you must draw a TV screen which is 40cm x 30cm in your notebook, but your notebook is too diminutive for you to draw a 40cm x 30cm TV screen. So you scale the size down, this phrase is very essential. You scale the size down that means you draw a smaller figure similar to the original thing. The actual TV size ratio is 40:30. Now you may draw a screen which is 4cm x 3cm and here the ratio is 4:3, or if you have a bigger notebook you may draw the TV screen which is 8cm x 6cm and here the ratio is 8:6.

In the first case, Why do we have to draw it in the ratio 4:3 why not 1:2.

Here is why the original size of the TV screen is 40 per 30 so if we reduce the ratio we will get 4 :3.  Same for the second case 8:6. So if you’ve observed, when we scale down the numbers we were maintaining the same relationship.

So 40:30,4:3,8:6 are equivalent ratios so 40:30, 4:3, 8:6 are in proportion.
we represent proportion ratos in this way.

40:30 : : 4:3 : : 8:6.
: : means ‘is the same as’.

here is an example to understand the use proportions
(The big picture shows the original size)
here in this example, the bigger image and the smaller image are in proportion.

in the big picture the hands, legs, hight and width was reduced proportionally To a small picture.

In the following figure, we have not reduced the original image (big image) proportionally.

Another example of proportions

Shown cases are not proportions.

Her in the above image, one student can read 1 book per 2 days and he can eat 5 pizzas per 10 days (1/2). Even though both are equal, but not comparing the same units. So these are not in proportion.
Note: Ratios are Proportion when having the same unit comparison.

In the above picture when the units are mismatched (1 book per 2 days ≠ 5 days per 10 books)  the ratios are not in proportion.

 

The above picture is a proportion because the unit is not mismatched.

Unit Rate

Did you know that  40mph (miles per hour) is also a ratio? why?
Remember any number can be written as a fraction by adding ‘/1’ so 40/1 is two numbers.  Well, 40 mph is the ratio 40 miles per 1 hour

and this type of ratio is a rate. A rate is noting but a ratio that involves a period of time, and if the bottom number is a 1 then it is called a unit rate.

Remember: A rate is not only speed it can be time or money.

Finding Unit Rate

A unit rate is really useful for comparing two things.

Car A🚗 is at a speed of 120 miles per 3 hours and Car B🚙. is going at a speed of 150 miles per 5 hours. which car is going faster? it is hard to compare which car is going faster, but you can do it by finding the unit rate.

1.  divide the numbers
   120/3 & 150/5
   120/3=40
   150/5=30
2. Now the unit rate of car A is 40mph and the unit rate of car B is 30.
   now it is easy to determine that Car A is faster than Car B.

Find a missing number in an equivalent ratio.

Cross Multiplying

Cross multiplying is used to find missing equivalent ratios. to know what cross multiplying is, here is an example.

ex.

A student who is a good reader can read 1book in 2 days. In how many days will he read 23 books in?

• •  list the ratio in the fraction form.
    

    ½ = 23/n
    

    Assume that ‘n’ is the unknown or missing value (23 per days)

• We imagine a cross symbol on the fractions
  
• then we multiply the pairs red cross 1xn & black cross 2×23
  

  1xn=2×23

  (we write ‘=’ because both ratios are in proportion)

• then we multiply the side with both numbers
  

  1xn = 46

• here If we move a ‘+’ equation to the right the equation will become a ‘ – ‘.
• equation opposite if the  ‘-‘  equation will be ‘+’.
• If we move an ‘x’ equation to the right the equation will become a ‘ ÷ ‘.
• the equation is ‘÷ then it will be  ‘ – ‘ equation.
• So now to find ‘n’ value, we will have to move 1 to another side then, the equation will become
  n = 46÷1
  n = 46
  
  
• now the equation is n=46 means he takes 46 days to complete 23 books.