Mathematics Short Tricks/ MATH RATIO AND PROPORTION

Mathematics Short Tricks/ MATH RATIO AND PROPORTION

 Mathematics Short Tricks – Ratio & Proportion

Ratio and Proportion are essential topics for competitive exams like SSC, Banking, GATE, CAT, etc. Here are some quick tricks & formulas to solve ratio and proportion problems FAST!

 What is Ratio?

A ratio compares two quantities and is written as:

$$\text{Ratio}=a:b$$

Example: If a class has 20 boys and 30 girls, the ratio of boys to girls is:

$$20:30=2:3$$

 Always simplify ratios by dividing by the HCF of both terms.

 What is Proportion?

A proportion states that two ratios are equal:

$$a:b=c:d$$

Here, b & c are mean terms, and a & d are extreme terms.

$$\text{Product of Means}=\text{Product of Extremes}$$

Example: Find x if $2:5=6:x$.

$$2\times x=5\times 6$$ x=302=15x = \frac{30}{2} = 15x=230​=15

Final Answer: $x=15$

 Trick to Solve Problems Faster

 Ratio Multiplication Trick

If two ratios are given:

$$a:b\text{and}c:d$$

To combine them into a single ratio:

$$(a\times c):(b\times d)$$

Example: Find the combined ratio of 2:3 and 4:5.

$$(2\times 4):(3\times 5)=8:15$$

Final Answer: $8:15$

 Trick for Finding a Number Based on Ratio

If x:y = m:n, then:

x=mm+n×Totalx = \frac{m}{m+n} \times \text{Total}x=m+nm​×Total y=nm+n×Totaly = \frac{n}{m+n} \times \text{Total}y=m+nn​×Total

Example: The ratio of A’s & B’s salary is 2:3, and their total salary is ₹5000. Find A’s and B’s salary.

A=22+3×5000=25×5000=2000A = \frac{2}{2+3} \times 5000 = \frac{2}{5} \times 5000 = 2000A=2+32​×5000=52​×5000=2000 B=35×5000=3000B = \frac{3}{5} \times 5000 = 3000B=53​×5000=3000

Final Answer: A = ₹2000, B = ₹3000

 How to Divide an Amount in a Given Ratio?

To divide ₹X in the ratio $a:b$, use:

First part=aa+b×X\text{First part} = \frac{a}{a+b} \times XFirst part=a+ba​×X Second part=ba+b×X\text{Second part} = \frac{b}{a+b} \times XSecond part=a+bb​×X

Example: ₹1000 is divided in the ratio 3:2.

First part=33+2×1000=35×1000=600\text{First part} = \frac{3}{3+2} \times 1000 = \frac{3}{5} \times 1000 = 600First part=3+23​×1000=53​×1000=600 Second part=25×1000=400\text{Second part} = \frac{2}{5} \times 1000 = 400Second part=52​×1000=400

Final Answer: ₹600 and ₹400

 Finding the New Ratio After an Increase/Decrease

If two quantities are increased/decreased by x% and y%, the new ratio is:

$$\text{New Ratio}=(a\times (100+x)):(b\times (100+y))$$

Example: The ratio of A’s and B’s salary is 4:5. If both are increased by 20% and 10%, find the new ratio.

$$\text{New Ratio}=(4\times 120):(5\times 110)=480:550$$

Simplify:

$$24:27.5$$

Multiply by 2 to remove decimals:

$$48:55$$

Final Answer: 48:55

 Quickest Way to Compare Two Ratios

To compare $a:b$ and $c:d$, cross multiply:
 If $a\times d>b\times c$ → $a:b$ is greater
 If $a\times d<b\times c$ → $c:d$ is greater

Example: Which is greater: 3:4 or 5:7?

$$3\times 7=21,4\times 5=20$$

Since 21 > 20, 3:4 is greater than 5:7.

 Mean Proportion Formula

The mean proportion between a and b is:

x=a×bx = \sqrt{a \times b}x=a×b​

Example: Find the mean proportion of 4 and 25.

x=4×25=100=10x = \sqrt{4 \times 25} = \sqrt{100} = 10x=4×25​=100​=10

Final Answer: 10

 Direct & Inverse Proportion

 Direct Proportion (More → More, Less → Less)

If x increases, y also increases, then:

x1:y1=x2:y2x_1:y_1 = x_2:y_2x1​:y1​=x2​:y2​

Example: If 5 workers do a job in 10 days, how many days for 10 workers?

$$5:10=10:x$$

Cross multiply:

$$5x=10\times 10\Rightarrow x=5$$

Final Answer: 5 days

 Inverse Proportion (More → Less, Less → More)

If x increases, y decreases, then:

x1×y1=x2×y2x_1 \times y_1 = x_2 \times y_2x1​×y1​=x2​×y2​

Example: If 8 men finish work in 12 days, how long will 16 men take?

$$8\times 12=16\times x$$ x=8×1216=6 daysx = \frac{8 \times 12}{16} = 6 \text{ days}x=168×12​=6 days

Final Answer: 6 days

 Summary of Tricks 

 Memorize basic ratio concepts
 Use cross multiplication to compare ratios
 Apply direct & inverse proportion rules
 Use proportion formula for quick calculations
 Use multiplication tricks to find new ratios

Master these tricks and solve any Ratio & Proportion question in seconds!

Would you like practice questions or PDFs for revision?

Mathematics Short Tricks/ MATH RATIO AND PROPORTION

Ratio and Proportion: Concepts, Solved Examples, & …

QUALITATIVE APTITUDE TRICKS & SHORTCUTS FOR

RATIO AND PROPORTION.pmd