Kit Library / Mathematics / Arithmetic
⚡ Topic Learning KitShared by a Veda learner · Generated with Veda AI
Veda Bites are swipeable micro-lessons — each one teaches exactly one idea. The bites below come from this kit — swipe through them in the app.
💡 Key Idea
Two ideas power half of arithmetic.
Ratio is a comparison; proportion is an equality of two ratios. Every proportion problem ultimately reduces to a ratio problem.
↳ Master ratios first — proportion is just two ratios set equal.
➗ Formula
One cross-multiplication solves most proportions.
In any proportion $a:b :: c:d$, the product of the extremes equals the product of the means. This single identity is the engine behind almost every proportion problem.
↳ Cross-multiply whenever a proportion has one unknown.
⭐ Important Fact
Ratios are pure numbers — no units.
↳ Convert to the same unit first; then only multiply/divide — never add/subtract.
⚙️ Process
Four steps crack any ratio question.
↳ The multiplier $k$ converts abstract ratios into solvable equations.
➗ Formula
Split any amount using one formula.
For ratio $a:b$, the total is split into $a+b$ equal parts. Each part $= \frac{\text{Total}}{a+b}$.
↳ Total parts = sum of ratio terms; each share = its term × one part.
⚖️ Comparison
Two proportions — opposite behaviours.
↳ Direct: ratio stays equal. Inverse: product stays equal.
⚠️ Common Mistake
Adding to both terms breaks the ratio.
↳ Ratio is preserved by multiplication/division, never by addition/subtraction.
LCM links two ratios in seconds.
↳ LCM of the common term is the fastest bridge between two ratios.
➗ Formula
Three named proportions, one pattern.
↳ All three come from $a:b = c:d$ — just plug in the knowns.
🌍 Real World
Business partners split profit by ratio.
In a partnership, profit is shared in the ratio of capital × time invested — not just capital.
↳ Partnership profit ratio = (capital × time) ratio.
🛠️ Application
Mixing liquids is a ratio game.
If milk:water = $3:2$ in 40 L, milk = $\frac{3}{5} \times 40 = 24$ L, water = 16 L.
↳ Every mixture question is a ratio split in disguise.
🎯 Exam Tip
Let $k$ do the heavy lifting.
↳ One multiplier $k$ turns ratio problems into simple linear equations.
🧠 Memory Trick
Extremes hug the outside.
In $a:b :: c:d$, the extremes ($a, d$) are on the outside; the means ($b, c$) are in the middle. Their products are equal.
↳ Outside × Outside = Inside × Inside.
⚠️ Common Mistake
Inverse means multiply, not divide.
↳ Direct → ratio equal. Inverse → product equal.
Fractions beat ratios in comparisons.
↳ Cross-multiplication compares ratios without decimals.
Ratio and proportion form the backbone of arithmetic problem-solving in SSC CGL — from partnership and mixture problems to time-speed-distance and data interpretation. This kit builds the conceptual clarity and shortcut…
Full explanation, worked examples & memory tricks in the app
All 18 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Q1. Which of the following correctly states the fundamental rule of proportion for $a:b :: c:d$?
BeginnerC · $ad = bc$
The product of extremes ($a \times d$) equals the product of means ($b \times c$).
Q2. A recipe uses flour and sugar in the ratio $3:2$. If 9 cups of flour are used, how many cups of sugar are needed?
BeginnerD · 6 cups
Direct proportion: $\dfrac{3}{2} = \dfrac{9}{x} \Rightarrow 3x = 18 \Rightarrow x = 6$.
Q3. In the proportion $4:7 :: 12:x$, what is the value of $x$?
BeginnerA · 21
$4x = 7 \times 12 = 84 \Rightarrow x = 21$.
Q4. If boys : girls in a class is $3:5$, what is the ratio of boys to the total number of students?
BeginnerC · $3:8$
Total = $3+5 = 8$ parts, so boys : total $= 3:8$.
Q5. Which of these is an example of inverse proportion?
BeginnerA · Number of workers and days to finish a fixed job
With fixed total work, more workers mean fewer days — the product is constant.
Q6. The fundamental property of proportion is $a:b :: c:d \iff ad = bc$ — the product of extremes equals the product of means.
BeginnerA · True
The fundamental property of proportion is $a:b :: c:d \iff ad = bc$ — the product of extremes equals the product of means.
Q7. The mean proportional between 4 and 25 is:
IntermediateB · 10
Mean proportional $= \sqrt{4 \times 25} = \sqrt{100} = 10$.
Q8. If $a:b = 2:3$ and $b:c = 4:5$, what is $a:b:c$?
IntermediateC · $8:12:15$
Make $b$ equal: $a:b = 8:12$ and $b:c = 12:15$, so $a:b:c = 8:12:15$.
Q9. ₹1,200 is divided among A, B and C in the ratio $2:3:5$. How much does B receive?
IntermediateD · ₹360
Total parts $= 10$, one part $= ₹120$, so B gets $3 \times 120 = ₹360$.
Q10. If 12 workers complete a job in 10 days, how many days will 15 workers take for the same job?
IntermediateC · 8 days
Inverse proportion: $12 \times 10 = 15 \times x \Rightarrow x = 8$ days.
Q11. The third proportional to 4 and 6 is:
IntermediateD · 9
Third proportional $= \dfrac{b^2}{a} = \dfrac{36}{4} = 9$.
Q12. A student writes 3 kg : 400 g as $3:400$. What is the correct ratio?
IntermediateB · $15:2$
Convert to the same unit: $3000:400 = 15:2$.
Q13. If $a:b = c:d$, which of the following is true by the invertendo property?
IntermediateB · $b:a = d:c$
Invertendo flips both ratios: $b:a = d:c$.
Q14. A map has a scale of $1:50{,}000$. If two towns are 4 cm apart on the map, what is the actual distance?
AdvancedA · 2 km
$4 \text{ cm} \times 50{,}000 = 200{,}000 \text{ cm} = 2 \text{ km}$.
Q15. A student claims that if $a:b = c:d$, then $a+b : b = c+d : d$. Which property justifies this?
AdvancedB · Componendo
Componendo adds the second term to the first in both ratios.
Q16. If $\dfrac{a+b}{a-b} = \dfrac{c+d}{c-d}$, which underlying proportion does this represent?
AdvancedD · $a:b = c:d$
This is the componendo–dividendo form, which holds when $a:b = c:d$.
Q17. If the ratio of boys to girls in a class is represented as $3:5$ and the total number of students is $40$, how many boys are there in the class?
AdvancedD · 18
The total parts in the ratio $3:5$ is $3 + 5 = 8$. Each part represents $40/8 = 5$ students. Therefore, boys = $3 imes 5 = 15$. The correct calculation shows $3:5$ means boys are $3$ parts out of $8$, resulting in $15$ boys.
Q18. A recipe requires flour and sugar in the ratio $4:1$. If you have 20 cups of flour, how many cups of sugar are needed to maintain the ratio?
AdvancedC · 5
The ratio $4:1$ indicates that for every 4 cups of flour, there is 1 cup of sugar. If you have 20 cups of flour, the amount of sugar needed is calculated as $20 \div 4 = 5$ cups.
20 flashcards in this kit — the app reviews them with spaced repetition so the right card returns on the right day.
Every game is built from this kit's own content — scores feed your mastery, so playing counts as studying.
Word Match
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True False
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Memory Match
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Flashcard Battle
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Speed Quiz
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Guess Term
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Sequence Builder
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Categorization
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Revision Battle
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