Ratio and Proportion Tricks: Formulas, Examples, FAQ

Ratio and Proportion Tricks and Shortcuts

Before coming to tricks, let’s have a quick recap of the basic concept of ratio and proportion.

When we compare two quantities using the method of division, it is known as a ratio. We use the ‘:’ sign to denote the ratio, represented as p:q.

Proportion is a comparison of two ratios. We use ‘::’ and ‘=’ signs to denote proportions. If two ratios p:q and r:s have equal value, they are said to be a proportion and represented as p:q::r:s.

Now let’s learn about the ratio and proportion tricks. 

• If p:q = r:s, then ps = qr.

Example: If 2:3 = 8:12 

Then, (2 × 12) = (8 × 3)

                   24  = 24, which satisfies the condition.

• If ratio p:q = ratio r:s, then (p/r) = (q/s). [Alternendo Rule]

Example: If 2:3 = 8:12 

Then, \frac{2}{8}=\frac{3}{12}

      ⇒ \frac{1}{4}=\frac{1}{4} , satisfying the condition.

• If ratio p:q = ratio r:s, then (q/p) = (s/r). [Invertendo Rule]

Example: If 2:3 = 8:12

Then \frac{3}{2}=\frac{12}{8}

     ⇒ 3/2 = 3/2, satisfying the condition.

• If p:q = r:s, then (p – r):(q – s). [Subtrahendo Rule]
  

• If p:q = r:s, then (p + r):(q + s). [Addendo Rule]

• If p:q = r:s, then (p + q) /q = (r + s) /s .[Dividendo Rule]

 Example: If 2:3 = 8:12

Then (2 + 3)/3 = (8 + 12)/12

     ⇒ \frac{5}{3}=\frac{20}{12}

     ⇒\frac{5}{3}=\frac{5}{3} , satisfying the condition.

• If p:q = r:s, then (p – q)/q = (r – s)/s. [Componendo Rule]

Example: If 2:3 = 8:12

Then\frac{(2-3)}{3}=\frac{(8-12)}{12} 

     ⇒\frac{-1}{3}=\frac{-4}{12} 

     ⇒ \frac{-1}{3}=\frac{-1}{3}, satisfying the condition.

•  If p:q = r:s, then (p+q)/ (p-q) = (r+s)/(r-s). [Componendo and Dividendo Rule]

Example: If 2:3 = 8:12

Then\frac{(2+3)}{(2-3)}=\frac{(8+12)}{(8-12)}

     ⇒ \frac{5}{(-1)}=\frac{20}{-4}

     ⇒ -5 = -5, satisfying the condition.

• If p:q = q:r, then p: r=\left(p^{2} / q^{2}\right).

Example: If 2:4 = 4:8

Then 2:8 = \frac{22}{42}22/42

     ⇒ \frac{2}{8}=\frac{4}{16}

     ⇒ \frac{1}{4}=\frac{1}{4} , satisfying the condition.

Some Important Properties of Ratio and Proportion

These properties are equally important along with the tricks explained above, So, we must know about these properties to solve problems related to this topic easily and quickly,

• If p:q = r:s, then

s is fourth proportional to p, q, and r.

r is the third proportion to p and q.

The mean proportion between p and q is √(pq)

• If (p:q)>(r:s) then  

  \frac{p}{q}>\frac{r}{s} 

• The compounded ratio of the ratios (p:q), (r:s), and (t:u) is (prt : qsu).
  
• If p:q is a ratio, then

           p^{2}: 1^{2} is the duplicate ratio,

          √p:√q is the sub-duplicate ratio, and

           p^{3}: q^{3} is the triplicate ratio.

Examples

1. If ratios (3:4) and (9:12) are equal in value, verify the Dividendo rule.

According to Dividendo rule, 

If p:q = r:s, then \frac{(p+q)} {q} = \frac{(r+s)} {s}

Here p = 3, q = 4, r = 9, and s = 12.

We have to verify \frac{(p+q)}{q}=\frac{(r+s)}{s}.

Substituting the values of p, q, r, and s.

\frac{(3+4)}{4} =\frac{(9+12)}{12}

⇒\frac{7}{4} = \frac{21}{12}

⇒\frac{7}{4} = \frac{7}{4}, which verifies the dividendo rule.

2. Find the compounded ratio of the ratios (2:3), (3:4), and (3:5).

We know that the compounded ratio of the ratios (p:q), (r:s), and (t:u) is (prt : qsu).

So, the compounded ratio of the ratios (2:3), (3:4), and (3:5)

= (2 × 3 × 3):(3 × 4 × 5)

= 18:60

= 3:10