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

Certainly! Mastering Ratio and Proportion is crucial for excelling in competitive exams like GATE, SSC, Banking, and others. Below are some effective tricks and shortcuts to enhance your problem-solving speed and accuracy.

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Key Concepts

• Ratio: A comparison between two quantities, expressed as a : b.
• Proportion: An equation stating that two ratios are equal, denoted as a : b :: c : d.(GeeksforGeeks, BYJU’S)

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Essential Tricks and Shortcuts

1. Mean Proportion:
   • If a : x = x : b, then x = √(a × b).
   • Example: Find the mean proportion between 4 and 9.
     • x = √(4 × 9) = √36 = 6
2. Third Proportion:
   • If a : b = b : c, then c = b² / a.
   • Example: Find the third proportion to 4 and 8.
     • c = 8² / 4 = 64 / 4 = 16
3. Fourth Proportion:
   • If a : b = c : d, then d = (b × c) / a.
   • Example: Find the fourth proportion to 3, 9, and 12.
     • d = (9 × 12) / 3 = 108 / 3 = 36
4. Componendo and Dividendo Rule:
   • If a / b = c / d, then:
     • (a + b) / (a - b) = (c + d) / (c - d)
   • Example: If a / b = 3 / 5, then:
     • (a + b) / (a - b) = (3 + 5) / (3 - 5) = 8 / (-2) = -4
5. Dividing a Quantity in a Given Ratio:
   • To divide amount A in the ratio m : n:
     • First part = (A × m) / (m + n)
     • Second part = (A × n) / (m + n)
   • Example: Divide 1800 in the ratio 2:3.
     • First part = (1800 × 2) / (2 + 3) = 3600 / 5 = 720
     • Second part = 1800 - 720 = 1080
6. Comparison of Ratios:
   • To compare a : b and c : d, convert both to fractions:
     • a / b and c / d
   • Compare the decimal values.
   • Example: Compare 3:4 and 5:7.
     • 3/4 = 0.75, 5/7 ≈ 0.714
     • Since 0.75 > 0.714, 3:4 > 5:7
7. Compound Ratio:
   • The compound ratio of a : b and c : d is (a × c) : (b × d).
   • Example: Find the compound ratio of 2:3 and 4:5.
     • (2 × 4) : (3 × 5) = 8 : 15
8. Duplicate, Triplicate, and Sub-Duplicate Ratios:
   • Duplicate Ratio: (a² : b²)
   • Triplicate Ratio: (a³ : b³)
   • Sub-Duplicate Ratio: (√a : √b)
   • Example: For 3:4:
     • Duplicate: 9:16
     • Triplicate: 27:64
     • Sub-Duplicate: √3:√4 ≈ 1.732:2

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Recommended Resources

• Guidely: Offers a comprehensive PDF on ratio and proportion tricks tailored for bank exams. (CareerRide, GeeksforGeeks, Guidely)
• GeeksforGeeks: Provides detailed explanations and examples on ratios and proportions. (Scribd)
• BYJU’S: Covers definitions, formulas, and solved examples to strengthen your understanding.

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Video Tutorials

For visual learners, the following video tutorials can be immensely helpful:

Ratio and Proportion – Shortcuts & Tricks for Placement Tests

Ratio and Proportion – Shortcut Tricks to Solve

Ratio and Proportion Word Problems – Math

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Feel free to reach out if you need further clarification or additional practice problems on any of these topics!