Ratio and Proportions Formula

Ratio is a representation of two quantities ‘a’ and ‘b’ having the same units. The ratio is defined as the fraction of two numbers a/b, which can be expressed as a : b in terms of their ratio.

If two numbers are a and b, then their ratio is 

a : b

Properties of Ratios:

• Two different ratios may represent the same value and are known as equivalent ratios.
• Dividing or multiplying a ratio with a constant gives back an equivalent ratio.
• If two ratios are equivalent, i.e. a : b = c : d, then we can write ad = bc by applying cross-multiplication.
• If a : b = c : d, then we can also write it as b : a = d : c.

Proportion formula is a representation of two equivalent ratios. This means, if the two ratios are equal, we say that they are proportional to each other.

If a ratio a:b is equivalent to c:d, that is a/b = c/d, then we say that a:b is proportional to c:d.

If a/b is proportional to c/d, then it can be expressed as,

a/b = c/d  or  a:b :: c:d

Properties of Proportions:

• If a : b = c : d, then we can say that (a + c) : (b + d), it is also called Addendo.
• If a : b = c : d, then we can say that (a – c) : (b – d), it is also called Subtrahendo.
• If a : b = c : d, then we can say that (a – b) : b = (c – d) : d, it is also known as Dividendo.
• If a : b = c : d, then we can say that (a + b) : b = (c + d) : d, it is also known as Componendo.
• If a : b = c : d, then we can say that a : c = b : d, it is also known as Alternendo.
• If a : b = c : d, then we can say that b : a = d : c, it is also called Invertendo.
• If a : b = c : d, then we can say that (a + b) : (a – b) = (c + d) : (c – d), it is also known as Componendo and Dividendo.
• If a is proportional to b, then it means a = kb where k is a constant.
• If a is inversely proportional to b, then a = k/b, where k is a constant.
• Dividing or multiplying a ratio by a certain number gives an equivalent ratio.

Sample Problems

Question 1. Is the ratio 5 : 10 proportional to 1 : 2?

Solution:

5 : 10 divided by 5 gives 1 : 2. Thus, they are same to each other. So we can say that 5 : 10 is proportional to 1 : 2.

Question 2. Given a constant k, such that k : 5 is proportional to 10 : 25. Find the value of k.

Solution:

Since k : 5 is proportional to 10 : 25, we can write,

k / 5 = 10 / 25

k = 10/25 × 5 = 2

So, the value of k is 2.

Question 3. Divide 100 into two parts such that they are proportional to 3 : 5.

Solution:

Let’s the value of two parts are 3k and 5k, where k is a constant.

Since the total sum of two parts is 100, we can write,

3k + 5k = 100

8k = 100

k = 12.5

So, the parts are 3k = 3 × 12.5 = 37.5 and 5k = 5 × 12.5 = 62.5

Question 4. If x² + 6y² = 5xy, then find the value of x/y.

Solution:

Given, x² + 6y² = 5xy. 

Dividing the equation by y², we get

(x/y)² + 6 = 5 (x/y)

Let’s x/y = t

So, we can write,

t² + 6 = 5t

t² – 5t + 6 = 0

(t – 2)(t – 3) = 0

t = 2 or t = 3

Since, t = x/y, we get

x/y = 2 or x/y = 3

Question 5. If a : b = c : d, then find the value of (a² + b²)/(c² + d²).

Solution:

Given, a/b = c/d. 

Squaring both sides, we get

a²/b² = c²/d²

a² = b²c²/d²

Putting the value of a² inside (a² + b²)/(c² + d²), we get

((b²c²/d²) + b²)/(c² + d²) =  b²(c² + d²)/(d² (c² + d²)) = b²/d²

Hence, our answer is b²/d² .

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