Ratio - Basics

The following skills will be learnt in this lesson on Ratio.

1. Ratio expresses a relation between two similar quantities.


2. A ratio can be written as a fraction


3. Terms in a ratio must be in same units


4. A Ratio does not change even though its terms may be divided or multiplied by a same number


5. A Ratio changes on adding to or subtracting from the terms of the ratio a same number


6. How to compare ratios

Consider the following:

The height of Eiffel tower is 324 m and the height of Liberty Statue is 93m.

Now, there are two ways to compare the heights of the tower and the statue

One is Difference and another is Division

The difference first:

In this method, we compare two things by finding how much more is one than the other.

Or, we find how much amount does one exceed the other by.

1. How much more tall is Eiffel tower than the liberty statue?

Answer:

Before finding, what equation is written for the expression below?

“p is how much more than q” (by what amount p exceeds q)

 p =? + q

So, for 324 is how much more than 93, write the equation:

324 = x + 93 i.e.  

x = 324 – 93 = 231

So, the height of the tower is 231m more than the height of the statue.

In simple words,

The tower is 231m taller than the statue.

And the other way is

Division

2. How many times is the height of the tower more than the height of the statue?

Answer:

The equation for

‘If x is n times y’ or ‘if x is n times more than y’, is

x = n × y

Therefore,

“324 is how many times 93” Or “324 is how many times more than 93” is the equation:

324 = n × 93.

Transpose 93 to the denominator in the R.H.S.

n = 324/93 = 3.48

(Round off 3.48 to 3.5)

The height of the tower is 3.5 times as much as the height of the statue.  

In short, the height of Eiffel tower is 3.5 times (more than) the height of liberty statue.

Caution:

Do not say:

The tower is 3.5 times the statue Or

The height of the tower is 3.5 times the statue

Because in the first expression above, the tower and statue are being compared which does not make much sense after all, and in the second, height is being compared to an object, which is odious.

Therefore, to stay free of any linguistic and semantic error, compare the heights of the two objects and say

The tower is 3.5 times as tall as the statue or

The height of the tower is 3.5 times that of the statue.

Let ‘x’ and ‘y’ denote the respective heights of the tower and the statue.

Then,

x = 3.5y Or

x/y = 3.5/1

The fractions x/y and 3.5/1 are also respectively denoted as

x: y and 3.5: 1

So, we have x: y = 3.5: 1

x: y is called the ratio of x to y and 3.5: 1 is called the ratio of 3.5 to 1

and therefore,

x: y = 3.5: 1 is defined as

‘the ratio of x to y is same as the ratio of 3.5 to 1’

Alternatively, we say

x: y as “x is to y”, so we have

x is to y is same as 3.5 is to 1.

The ratio x: y = 3.5: 1 (in words say it as “x is to y is same as 3.5 is to 1) can be expressed as a

A ratio expresses a relation between two similar quantities.

Example:

If my house is twice as far from office as from gym, then write a relation between the distances.

Let ‘x’ be the distance between house and gym

And 'y’ be the distance between house and office, then

Distance of house from office = 2 × (distance between house and office).

i.e. y = 2x

You can also write y/x = 2.

y/x = 2 can be described as a ratio as

The ratio of the distance of my house from office to that from gym is 2 to 1

In simple words, it is

My house is twice as far from office as from gym

Alternatively, in fraction terms, it is

The distance of gym from my house is half of that from office.

Terms of a ratio must be in same units

Suppose a bottle contains 2 litres and a can 2 gallons.  

Is the ratio of water contained by the two same?

Is it 2: 2 i.e. 1: 1

No

The capacities are expressed in two different units.

To define a ratio on them, first convert gallons into litres.

Now, 1 gallon is 3.78 litres roughly.

(You can as well round off 3.78 to 3.8)

So, 2 gallons are 2 × 3.78 = 7.56 litres.

Round off 7.56 to 7.6

Now, express the volumes using a ratio as below

2 gallons to 2 litres = 7.56 litres: 2 litres, i.e. the ratio is

7.56: 2 = 3.8: 1

Therefore, the can contains 3.8 times as much as the bottle does, Or

The ratio of capacity of the can to that of the bottle is 3.8 to 1,

Or the ratio of volume of can to that of bottle is same as the ratio of 3.8 to 1.

CAUTION:

Ratios do not have units.

For example,

The ratio of 2 cms to 3 cms is 2: 3 and not 2: 3 cms.

Ratio only compares, it is not a quantity itself that is expressed in units.

Express ratio on the following:

 1. 5 miles to 5 kilometres

The ratio of 5miles to 5 kms is not the same as 5: 5, i.e. 1: 1.

Because one mile is as many as 8/5 kms,

So, 5 miles make 5 × 8/5 = 8 kms.

Therefore,

The ratio of 5 miles to 5 kms is same as the ratio of 8 kms to 5 kms, which is 8: 5

The ratio of 8 to 5 is 8: 5

2. One metric tonne to one quintal

One metric tonne = 1000kg and one quintal = 100kg

So the ratio of one tonne to one quintal is same as that of 1000kg to 100kg. Now,

The ratio of 1000 to 100 is

1000: 100 = 10: 1

3. One day to one hour

One day is 24 hours long.

So, the ratio of one day to one hour is

24 hours: 1 hour = 24: 1

4. 9 yards to 10 metres.

1 yard = 3 feet, 9 yards = 3 × 9 = 27 feet, and

1 metre = 3.3 feet (roughly), 10 metres = 10 × 3.3 = 33 feet.

So, 9 yards: 10 metres = 27 feet: 33 feet = 27: 33

In the ratio 27: 33,   

The two terms 27 and 33 have a common factor, namely 3.

Divide each term by 3 to simplify the ratio.

Practically speaking, strike out this common factor from the two terms 27 and 33 in the ratio 27: 33 as below

Ratio does not change even though its terms are divided or multiplied by a same number

Example 1:

What is the ratio of angle of a straight line to one complete rotation?

Angle of a straight line is 180⁰ and one full rotation traverses 360⁰

Therefore the ratio of 180⁰: 360⁰ is

180⁰: 360⁰ = 1: 2

But,

  =

,

which is also the ratio of 90⁰: 180⁰ which is 1: 2

Therefore, the ratio of 90 deg to 180 deg is same as that of 180 deg to 360 deg.

Therefore, if a ratio is p: q, then

p: q = 2p: 2q,

in fact p: q = np: nq, where n is any number.

An important note:

Ratio is same, though terms may be any.

In the ratio 1: 2, 1 and 2 are terms, not necessarily the two exact numbers.

Though numbers may vary, still their ratio need not change

Observe

Example 1:

Ratio of 1 and 2 is 1: 2,

Example 2:

Ratio of 2 and 4 is 2: 4 = 2×1: 2×4 = 1: 2

Example 3:

Ratio of 100 and 200 is 100: 200 = 100×1: 100 × 2 = 1: 2

In the three examples above, between different numbers, there can be still a same ratio.

This leads to the following important note:

If ratio of two numbers is a: b, then the numbers are not necessarily exactly a and b.

To find two numbers whose ratio is a: b, then let

One number be x and the other number be y.

Since x: y = a: b, take

x = p × a and y = p × b

where p is a common factor such as 2 and 100 in the 2nd and 3rd examples above.

Example:

The ratio of two numbers is 3: 5. Find two pairs of such numbers.

Answer:

Let the two numbers be x and y,

Now x: y = 3: 5

From the above note,

x = 3p and y = 5p,

where p is a common factor of x and y and any number.

You can choose to write integers such as 2, 3 ... for p to find the various numbers as follows:

x = 3 ×1, 3 × 2, and

y = 5 × 1, 5 × 2

So, among many, two pairs of numbers for x and y are

3, 5 and 6, 10

How to compare ratios

To compare ratios, express them as fractions and apply methods for comparing fractions.

Example:

Which of the two ratios below is greater?

3: 4 or 5: 6

Answer:

Express both ratios as fractions first  

3: 4 is 3/4 and 5: 6 is 5/6

Now,

Compare 3/4 and 5/6

Is 3/4 > 5/6

Cross multiply:

Is 3 × 6 > 5 × 4?

Is 18 > 20, No.

Since 18 < 20, we conclude

3/4 < 5/6

Though there are other methods for comparing fractions, but they are not as easy and fast as the method of cross multiplying.

For example:

Is 3/4 < 5/6

Find L.C.M. of denominators 4 and 6 and multiply it to both the fractions.

L.C.M. of 4 and 6 is 24. Now, multiply each fraction with 24 and compare the resulting products:

(3/4) ×24 = 6 and (5/6) × 24 = 20

If 18 < 20, Yes.

Therefore, 3/4 < 5/6

Another cumbersome method is to make numerators same and compare denominators. The fraction with lesser denominator is the greater one.

Ratio changes on adding to or subtracting from the terms of the ratio a same number

Consider the ratio 2: 3

Add 1 to each term. The new ratio is 3: 4

Is 2: 3 same as 3: 4?

That is a question on

How to compare two ratios?

To compare ratios, write them as fractions and compare fractions.

The ratio 2: 3 is the fraction 2/3, while

The ratio 3: 4 is the fraction 3/4

Now compare fractions 2/3 and 3/4

Cross-mult iply the two fractions  

So, the two fractions 2/3 and 3/4 are not the same, and

Therefore, their equivalent ratios 2: 3 and 3: 4 are also not same.

Note:

Just like adding, even subtracting a same number from the two terms will change the ratio

Example:

The ratio 4: 5 is not same as the ratio 3: 4