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A circular copper plate is heated, and its radius increases by 3.2%. What is the approximate percentage increase in its area?
तांबे की एक वृत्ताकार प्लेट को गर्म किया जाता है, और इसकी त्रिज्या में 3.2% की वृद्धि होती है। इसके क्षेत्रफल में लगभग कितने प्रतिशत की वृद्धि होगी?
(A) (a) 6.40%
(B) (b) 6.50%
(C) (c) 6.64%
(D) (d) 7.04%
✅ Answer & Explanation
Sahi jawab: B) (b) 6.50%Explanation: Method 1 (Successive Percentage Formula): Circle ka area $= \pi r^2$, yaani $\text{Area} \propto r^2$. Radius me do baar change count hoga. Net Change $= a + b + \frac{ab}{100}$ Yahan $a = 3.2\%$ aur $b = 3.2\%$: $= 3.2 + 3.2 + \frac{3.2 \times 3.2}{100} = 6.4 + 0.1024 = 6.5024\% \approx 6.50\%$. Atah sahi vikalp (b) 6.50% hai. Method 2 (100-Base Method): Initial area $= 100$ maan lo. Radius factor $= 1 + \frac{3.2}{100} = 1.032$. New Area $= 100 \times (1.032)^2 = 100 \times 1.065024 = 106.5024$. Area me percentage badhav $= 106.5024 - 100 \approx 6.50\%$. Atah sahi uttar (b) hai.
If the surface area of a sphere increases by 96%, by what approximate percent does the radius increase?
यदि एक गोले का पृष्ठीय क्षेत्रफल 96% बढ़ जाता है, तो इसकी त्रिज्या में कितने प्रतिशत की वृद्धि होगी?
(A) (a) 35%
(B) (b) 40%
(C) (c) 44%
(D) (d) 48%
✅ Answer & Explanation
Sahi jawab: B) (b) 40%Explanation: Method 1 (Ratio Method): Sphere ka surface area $= 4\pi r^2 \implies \text{Area} \propto r^2$. Area $96\%$ badh gaya: Initial Area : New Area $= 100 : (100 + 96) = 100 : 196$. Radius ka ratio lene ke liye square root nikaalo: Initial Radius : New Radius $= \sqrt{100} : \sqrt{196} = 10 : 14 = 5 : 7$. Radius me badhav $= 7 - 5 = 2$ units (5 ke base par). Percentage increase $= \frac{2}{5} \times 100\% = 40\%$. Atah sahi vikalp (b) 40% hai. Method 2 (Successive Check Method): Agar radius $40\%$ badhegi, toh successive rule se: $= 40 + 40 + \frac{40 \times 40}{100} = 80 + 16 = 96\%$. Yeh question ke data se perfectly match karta hai, isliye answer 40% hai. Atah sahi uttar (b) hai.
If the radius of a sphere is increased by 6%, then by what percentage will the surface area of the sphere increase?
यदि एक गोले की त्रिज्या में 6% की वृद्धि की जाती है, तो गोले के पृष्ठीय क्षेत्रफल में कितने प्रतिशत की वृद्धि होगी?
(A) (a) 12.00%
(B) (b) 12.36%
(C) (c) 12.72%
(D) (d) 13.16%
✅ Answer & Explanation
Sahi jawab: B) (b) 12.36%Explanation: Method 1 (Successive Formula): Surface Area $\propto r^2$. Formula: $x + y + \frac{xy}{100}$ Yahan $x = 6\%$ aur $y = 6\%$: $= 6 + 6 + \frac{6 \times 6}{100} = 12 + 0.36 = 12.36\%$. Atah sahi vikalp (b) 12.36% hai. Method 2 (Ratio Method): $6\% = \frac{3}{50}$. Radius ratio $= 50 : 53$. Area ratio $= 50^2 : 53^2 = 2500 : 2809$. Increase $= 2809 - 2500 = 309$ units. Percentage increase $= \frac{309}{2500} \times 100\% = \frac{309}{25}\% = 12.36\%$. Atah sahi uttar (b) hai.
The radius of a sphere is increased by 15%. What is the increase percent in its surface area?
एक गोले की त्रिज्या में 15% की वृद्धि की जाती है। इसके पृष्ठीय क्षेत्रफल में कितने प्रतिशत की वृद्धि होगी?
(A) (a) 30.00%
(B) (b) 31.25%
(C) (c) 32.25%
(D) (d) 33.75%
✅ Answer & Explanation
Sahi jawab: C) (c) 32.25%Explanation: Method 1 (Ratio Method): $15\% = \frac{3}{20}$. Radius ratio $= 20 : (20 + 3) = 20 : 23$. Surface Area $\propto r^2$. Area ratio $= 20^2 : 23^2 = 400 : 529$. Increase in area $= 529 - 400 = 129$ units. Percentage increase $= \frac{129}{400} \times 100\% = \frac{129}{4}\% = 32.25\%$. Atah sahi vikalp (c) 32.25% hai. Method 2 (Successive Formula): Net Change $= 15 + 15 + \frac{15 \times 15}{100} = 30 + 2.25 = 32.25\%$. Atah sahi uttar (c) hai.
If the side of a square is increased by 8%, then the percentage increase in its area will be:
यदि एक वर्ग की भुजा में 8% की वृद्धि की जाती है, तो उसके क्षेत्रफल में प्रतिशत वृद्धि होगी:
(A) (a) 16.00%
(B) (b) 16.48%
(C) (c) 16.64%
(D) (d) 17.28%
✅ Answer & Explanation
Sahi jawab: C) (c) 16.64%Explanation: Method 1 (Ratio Method): $8\% = \frac{2}{25}$. Side ratio $= 25 : 27$. Square area $= \text{side}^2$. Area ratio $= 25^2 : 27^2 = 625 : 729$. Increase in area $= 729 - 625 = 104$ units. Percentage increase $= \frac{104}{625} \times 100\% = \frac{416}{25}\% = 16.64\%$. Atah sahi vikalp (c) 16.64% hai. Method 2 (Successive Formula): Net Increase $= 8 + 8 + \frac{8 \times 8}{100} = 16 + 0.64 = 16.64\%$. Atah sahi uttar (c) hai.
The breadth of a rectangular field is decreased by 12% and its length is increased by 25%. Find the percentage change in the area of the field.
एक आयताकार मैदान की चौड़ाई में 12% की कमी की जाती है और इसकी लंबाई में 25% की वृद्धि की जाती है। मैदान के क्षेत्रफल में प्रतिशत परिवर्तन ज्ञात कीजिए।
(A) (a) 10% increase
(B) (b) 10% decrease
(C) (c) 13% increase
(D) (d) 13% decrease
✅ Answer & Explanation
Sahi jawab: A) (a) 10% increaseExplanation: Method 1 (Ratio Method): Length: $25\% = \frac{1}{4} \implies 4 : 5$. Breadth: $12\% = \frac{3}{25} \implies 25 : 22$. Area $=$ Length $\times$ Breadth. Initial Area : New Area $= (4 \times 25) : (5 \times 22) = 100 : 110$. Area 100 se badhkar 110 hua, yaani seedhe $10\%$ increase. Atah sahi vikalp (a) 10% increase hai. Method 2 (Successive Formula): Formula: $x + y + \frac{xy}{100}$ Yahan $x = +25$ aur $y = -12$: $= 25 - 12 + \frac{25 \times (-12)}{100} = 13 - 3 = +10\%$ (10% increase). Atah sahi uttar (a) hai.
The radius of a circle is increased in such a way that its circumference increases by 11%. As a result, the area of the circle also increases by:
एक वृत्त की त्रिज्या इस प्रकार बढ़ाई जाती है कि उसकी परिधि में 11% की वृद्धि होती है। परिणामस्वरूप, वृत्त के क्षेत्रफल में कितने प्रतिशत की वृद्धि होगी?
(A) (a) 22.00%
(B) (b) 22.81%
(C) (c) 23.21%
(D) (d) 24.11%
✅ Answer & Explanation
Sahi jawab: C) (c) 23.21%Explanation: Method 1 (Successive Formula): Circumference $C = 2\pi r$, isliye circumference aur radius dono me same rate ($11\%$) se badhav hoga. Area $\propto r^2$ hone ki wajah se successive change lagega: $= 11 + 11 + \frac{11 \times 11}{100} = 22 + 1.21 = 23.21\%$. Atah sahi vikalp (c) 23.21% hai. Method 2 (100-Base Method): Initial area $= 100$. New Area $= 100 \times \left(\frac{111}{100}\right) \times \left(\frac{111}{100}\right) = \frac{12321}{100} = 123.21$. Increase over base $100 = 123.21 - 100 = 23.21\%$. Atah sahi uttar (c) hai.
The length of a rectangle is increased by 15% and its breadth is increased by 10%. Find the percentage increase in the area.
एक आयत की लंबाई में 15% की वृद्धि की जाती है और इसकी चौड़ाई में 10% की वृद्धि की जाती है। क्षेत्रफल में प्रतिशत वृद्धि ज्ञात कीजिए।
(A) (a) 25.0%
(B) (b) 26.5%
(C) (c) 27.2%
(D) (d) 28.0%
✅ Answer & Explanation
Sahi jawab: B) (b) 26.5%Explanation: Method 1 (Successive Formula): Formula: $x + y + \frac{xy}{100}$ Yahan $x = +15\%$ aur $y = +10\%$: $= 15 + 10 + \frac{15 \times 10}{100} = 25 + 1.5 = 26.5\%$. Atah sahi vikalp (b) 26.5% hai. Method 2 (Ratio Method): Length: $15\% = \frac{3}{20} \implies 20 : 23$. Breadth: $10\% = \frac{1}{10} \implies 10 : 11$. Area ratio $= (20 \times 10) : (23 \times 11) = 200 : 253$. Base ko 100 banane ke liye 2 se divide karo: $100 : 126.5$. Net increase $= 126.5 - 100 = 26.5\%$. Atah sahi uttar (b) hai.
The radius of a circular wire is reduced by 5%. What is the percentage decrease in its cross-sectional area?
एक वृत्ताकार तार की त्रिज्या में 5% की कमी की जाती है। इसके अनुप्रस्थ काट (cross-sectional) के क्षेत्रफल में कितने प्रतिशत की कमी होगी?
(A) (a) 9.75%
(B) (b) 10.00%
(C) (c) 10.25%
(D) (d) 10.50%
✅ Answer & Explanation
Sahi jawab: A) (a) 9.75%Explanation: Method 1 (Successive Formula): Area $\propto r^2$. Yahan $x = -5\%$ aur $y = -5\%$: Net Change $= -5 - 5 + \frac{(-5) \times (-5)}{100} = -10 + 0.25 = -9.75\%$ (9.75% decrease). Atah sahi vikalp (a) 9.75% hai. Method 2 (Ratio Method): $5\% = \frac{1}{20}$. Radius ratio $= 20 : 19$. Area ratio $= 20^2 : 19^2 = 400 : 361$. Area me kami $= 400 - 361 = 39$ units. Percentage decrease $= \frac{39}{400} \times 100\% = \frac{39}{4}\% = 9.75\%$. Atah sahi uttar (a) hai.
If all three dimensions of a cuboid are increased by 15%, the volume increases by what percentage?
यदि एक घनाभ की तीनों विमाओं में 15% की वृद्धि की जाती है, तो आयतन में कितने प्रतिशत की वृद्धि होगी?
(A) (a) 45.00%
(B) (b) 50.25%
(C) (c) 52.09%
(D) (d) 55.45%
✅ Answer & Explanation
Sahi jawab: C) (c) 52.09%Explanation: Method 1 (Ratio Method): $15\% = \frac{3}{20}$. Teeno dimensions ka ratio $= 20 : 23$. Volume $= l \times b \times h$. Initial Volume : New Volume $= 20^3 : 23^3 = 8000 : 12167$. Volume me increase $= 12167 - 8000 = 4167$ units. Percentage increase $= \frac{4167}{8000} \times 100\% = \frac{4167}{80}\% = 52.0875\% \approx 52.09\%$. Atah sahi vikalp (c) 52.09% hai. Method 2 (100-Base Method): Initial Volume $= 100$. Pehle do dimensions ka net increase: $15 + 15 + \frac{15 \times 15}{100} = 30 + 2.25 = 32.25\%$. Ab $32.25\%$ aur teesri dimension ($15\%$) ka net increase: $= 32.25 + 15 + \frac{32.25 \times 15}{100} = 47.25 + 4.8375 = 52.0875\% \approx 52.09\%$. Atah sahi uttar (c) hai.
The base of a triangle is increased by 25%. By what percentage should its height be increased so that the area increases by 75%?
एक त्रिभुज के आधार में 25% की वृद्धि की जाती है। इसकी ऊंचाई में कितने प्रतिशत की वृद्धि की जानी चाहिए ताकि क्षेत्रफल में 75% की वृद्धि हो जाए?
(A) (a) 35%
(B) (b) 40%
(C) (c) 45%
(D) (d) 50%
✅ Answer & Explanation
Sahi jawab: B) (b) 40%Explanation: Method 1 (Ratio Method): Area of triangle $= \frac{1}{2} \times \text{base} \times \text{height} \implies \text{Height} = \frac{\text{Area}}{\text{Base}}$. Base ratio: $25\% = \frac{1}{4} \implies 4 : 5$. Area ratio: $75\% = \frac{3}{4} \implies 4 : 7$. Height ratio $= \frac{4}{4} : \frac{7}{5} = 1 : 1.4 = 5 : 7$. Height me increase $= 7 - 5 = 2$ units (5 ke base par). Percentage increase $= \frac{2}{5} \times 100\% = 40\%$. Atah sahi vikalp (b) 40% hai. Method 2 (100-Base Method): Initial Area $= 100$. Base $25\%$ badhne ke baad area ho jayega $= 125$. Target area $= 175$ (total increase $75\%$). Height ki wajah se required increase $= 175 - 125 = 50$. 125 ke base par percentage increase: $= \frac{50}{125} \times 100\% = \frac{2}{5} \times 100\% = 40\%$. Atah sahi uttar (b) hai.
If the radius of a sphere is increased by 12.5%, then by what per cent would its surface area increase (correct to one decimal place)?
यदि एक गोले की त्रिज्या में 12.5% की वृद्धि की जाती है, तो इसके पृष्ठीय क्षेत्रफल में कितने प्रतिशत की वृद्धि होगी?
(A) (a) 25.0%
(B) (b) 26.6%
(C) (c) 27.2%
(D) (d) 28.5%
✅ Answer & Explanation
Sahi jawab: B) (b) 26.6%Explanation: Method 1 (Ratio Method): $12.5\% = \frac{1}{8}$. Radius ka ratio $= 8 : (8 + 1) = 8 : 9$. Surface Area $\propto r^2$. Area ratio $= 8^2 : 9^2 = 64 : 81$. Area me increase $= 81 - 64 = 17$ units. Percentage increase $= \frac{17}{64} \times 100\% = \frac{425}{16}\% \approx 26.56\% \approx 26.6\%$. Atah sahi vikalp (b) 26.6% hai. Method 2 (Successive Formula): Net increase $= 12.5 + 12.5 + \frac{12.5 \times 12.5}{100} = 25 + 1.5625 = 26.5625\% \approx 26.6\%$. Atah sahi uttar (b) hai.
Find the percentage increase in the surface area of a cube if each side is made 5 times its original length.
यदि एक घन की प्रत्येक भुजा को उसकी मूल लंबाई का 5 गुना कर दिया जाए, तो उसके पृष्ठीय क्षेत्रफल में प्रतिशत वृद्धि ज्ञात कीजिए।
(A) (a) 2000%
(B) (b) 2400%
(C) (c) 2500%
(D) (d) 1600%
✅ Answer & Explanation
Sahi jawab: B) (b) 2400%Explanation: Method 1 (Ratio Method): Cube ka surface area $= 6a^2 \implies \text{Area} \propto a^2$. Side ratio $= 1 : 5$. Area ratio $= 1^2 : 5^2 = 1 : 25$. Area me increase $= 25 - 1 = 24$ units (1 ke base par). Percentage increase $= \frac{24}{1} \times 100\% = 2400\%$. Atah sahi vikalp (b) 2400% hai. Method 2 (100-Base Method): Initial surface area $= 100$. Side 5 guna hone par naya area $= 100 \times 5^2 = 100 \times 25 = 2500$. Increase $= 2500 - 100 = 2400\%$. Atah sahi uttar (b) hai.
If the radius of a sphere is made 4 times its initial value, then its surface area will be increased by:
यदि किसी गोले की त्रिज्या को उसके प्रारंभिक मान का 4 गुना कर दिया जाए, तो उसके पृष्ठीय क्षेत्रफल में कितने प्रतिशत की वृद्धि होगी?
(A) (a) 1200%
(B) (b) 1500%
(C) (c) 1600%
(D) (d) 800%
✅ Answer & Explanation
Sahi jawab: B) (b) 1500%Explanation: Method 1 (Ratio Method): Surface Area $\propto r^2$. Radius ratio $= 1 : 4$. Area ratio $= 1^2 : 4^2 = 1 : 16$. Area me increase $= 16 - 1 = 15$ units. Percentage increase $= \frac{15}{1} \times 100\% = 1500\%$. Atah sahi vikalp (b) 1500% hai. Method 2 (100-Base Method): Initial surface area $= 100$. Radius 4 guna hone par naya area $= 100 \times 4^2 = 100 \times 16 = 1600$. Increase in area $= 1600 - 100 = 1500\%$. Atah sahi uttar (b) hai.
If the area of a square is decreased by 51%, then the diagonal of the square is decreased by:
यदि एक वर्ग के क्षेत्रफल में 51% की कमी होती है, तो वर्ग के विकर्ण में कितने प्रतिशत की कमी होगी?
(A) (a) 25.5%
(B) (b) 30%
(C) (c) 28%
(D) (d) 32%
✅ Answer & Explanation
Sahi jawab: B) (b) 30%Explanation: Method 1 (Ratio Method): Square ka area $= \frac{d^2}{2} \implies \text{Area} \propto d^2$. Area $51\%$ kam ho gaya: Initial Area : New Area $= 100 : (100 - 51) = 100 : 49$. Diagonal ($d$) ka ratio nikalne ke liye square root lenge: Initial Diagonal : New Diagonal $= \sqrt{100} : \sqrt{49} = 10 : 7$. Diagonal me kami $= 10 - 7 = 3$ units (10 ke base par). Percentage decrease $= \frac{3}{10} \times 100\% = 30\%$. Atah sahi vikalp (b) 30% hai. Method 2 (Successive Check Method): Agar diagonal $30\%$ decrease hota hai: Area reduction $= -30 - 30 + \frac{(-30)(-30)}{100} = -60 + 9 = -51\%$. Yeh question ke diye gaye data se match karta hai, isliye answer 30% hoga. Atah sahi uttar (b) hai.
If the radius of a sphere is increased by 15%, then the volume will be increased by approximately:
यदि एक गोले की त्रिज्या में 15% की वृद्धि की जाती है, तो आयतन में लगभग कितने प्रतिशत की वृद्धि होगी?
(A) (a) 45.0%
(B) (b) 50.5%
(C) (c) 52.1%
(D) (d) 55.2%
✅ Answer & Explanation
Sahi jawab: C) (c) 52.1%Explanation: Method 1 (Ratio Method): Volume of sphere $= \frac{4}{3}\pi r^3 \implies \text{Volume} \propto r^3$. $15\% = \frac{3}{20}$. Radius ratio $= 20 : 23$. Volume ratio $= 20^3 : 23^3 = 8000 : 12167$. Increase in volume $= 12167 - 8000 = 4167$ units. Percentage increase $= \frac{4167}{8000} \times 100\% = \frac{4167}{80}\% = 52.0875\% \approx 52.1\%$. Atah sahi vikalp (c) 52.1% hai. Method 2 (100-Base Method): Initial Volume $= 100$. 1st step: $15 + 15 + \frac{15 \times 15}{100} = 32.25\%$. 2nd step: $32.25 + 15 + \frac{32.25 \times 15}{100} = 47.25 + 4.8375 = 52.0875\% \approx 52.1\%$. Atah sahi uttar (c) hai.
If the height of a cylinder is increased by 20% and its radius is decreased by 10%, what is the percentage change in its volume?
यदि एक बेलन की ऊंचाई में 20% की वृद्धि की जाती है और इसकी त्रिज्या में 10% की कमी की जाती है, तो इसके आयतन में कितने प्रतिशत का परिवर्तन होगा?
(A) (a) 2.8% decrease
(B) (b) 2.8% increase
(C) (c) 4.2% increase
(D) (d) 4.2% decrease
✅ Answer & Explanation
Sahi jawab: A) (a) 2.8% decreaseExplanation: Method 1 (Ratio Method): Volume of cylinder $= \pi r^2 h \implies \text{Volume} \propto r \times r \times h$. Radius ($10\%$ decrease $= \frac{1}{10}$): ratio $= 10 : 9$. Height ($20\%$ increase $= \frac{1}{5}$): ratio $= 5 : 6$. Initial Volume : New Volume $= (10^2 \times 5) : (9^2 \times 6) = 500 : 486$. Volume me kami $= 500 - 486 = 14$ units (500 ke base par). Percentage decrease $= \frac{14}{500} \times 100\% = 2.8\%$ decrease. Atah sahi vikalp (a) 2.8% decrease hai. Method 2 (100-Base Method): Radius me do baar $10\%$ kami ka net effect: $-10 - 10 + \frac{(-10)(-10)}{100} = -19\%$. Ab $-19\%$ aur height ke $+20\%$ ka net change: $= -19 + 20 + \frac{(-19) \times 20}{100} = 1 - 3.8 = -2.8\%$ (yaani 2.8% decrease). Atah sahi uttar (a) hai.
If the radius of a sphere is increased by 30%, then find the percentage increase in its volume.
यदि एक गोले की त्रिज्या में 30% की वृद्धि की जाती है, तो उसके आयतन में प्रतिशत वृद्धि ज्ञात कीजिए।
(A) (a) 119.7%
(B) (b) 120.5%
(C) (c) 115.8%
(D) (d) 125.0%
✅ Answer & Explanation
Sahi jawab: A) (a) 119.7%Explanation: Method 1 (Ratio Method): Sphere ka volume $V = \frac{4}{3}\pi r^3 \implies V \propto r^3$. Radius badhne ki dar $= 30\% = \frac{3}{10}$. Initial Radius : New Radius $= 10 : 13$. Volume ratio $= 10^3 : 13^3 = 1000 : 2197$. Volume me increase $= 2197 - 1000 = 1197$ units. Percentage increase $= \frac{1197}{1000} \times 100\% = 119.7\%$. Atah sahi vikalp (a) 119.7% hai. Method 2 (100-Base Method): Initial volume ko 100 maan lo. Pehle do radius change ka net: $30 + 30 + \frac{30 \times 30}{100} = 60 + 9 = 69\%$. Ab $69\%$ aur teesre $30\%$ ka net change: $= 69 + 30 + \frac{69 \times 30}{100} = 99 + 20.7 = 119.7\%$. Atah sahi uttar (a) hai.
If the length and breadth of a cuboid are increased by 5% and 10%, respectively, while its height remains constant, then the percentage increase in its volume is:
यदि एक घनाभ की लंबाई और चौड़ाई में क्रमशः 5% और 10% की वृद्धि की जाती है, जबकि उसकी ऊंचाई अपरिवर्तित रहती है, तो उसके आयतन में प्रतिशत वृद्धि क्या होगी?
(A) (a) 15.00%
(B) (b) 15.50%
(C) (c) 16.00%
(D) (d) 16.50%
✅ Answer & Explanation
Sahi jawab: B) (b) 15.50%Explanation: Method 1 (Ratio Method): Height constant hai, isliye Volume $\propto \text{Length} \times \text{Breadth}$. Length: $5\% = \frac{1}{20} \implies 20 : 21$. Breadth: $10\% = \frac{1}{10} \implies 10 : 11$. Volume ratio $= (20 \times 10) : (21 \times 11) = 200 : 231$. Base ko 100 banane ke liye 2 se divide karo: $100 : 115.5$. Volume me increase $= 115.5 - 100 = 15.5\%$. Atah sahi vikalp (b) 15.50% hai. Method 2 (Successive Formula): Net Change $= x + y + \frac{xy}{100}$ Yahan $x = 5\%$ aur $y = 10\%$: $= 5 + 10 + \frac{5 \times 10}{100} = 15 + 0.5 = 15.50\%$. Atah sahi uttar (b) hai.
A metallic conical vessel with base radius 12 cm and height 18 cm is uniformly heated, increasing all its dimensions by 20%. By how much percent does the volume increase?
12 सेमी आधार त्रिज्या और 18 सेमी ऊंचाई वाले एक शंक्वाकार धात्विक बर्तन को समान रूप से गर्म किया जाता है, जिससे इसकी सभी विमाओं में 20% की वृद्धि होती है। इसके आयतन में कितने प्रतिशत की वृद्धि होगी?
(A) (a) 60.0%
(B) (b) 68.4%
(C) (c) 72.8%
(D) (d) 80.0%
✅ Answer & Explanation
Sahi jawab: C) (c) 72.8%Explanation: Method 1 (Ratio Method): Cone ka volume $V = \frac{1}{3}\pi r^2 h$. Sabhi dimensions $20\%$ badh rahi hain, isliye actual measurements (12 cm aur 18 cm) par calculation depend nahi karegi. $20\% = \frac{1}{5} \implies$ Each dimension ratio $= 5 : 6$. Volume ratio $= 5^3 : 6^3 = 125 : 216$. Volume me increase $= 216 - 125 = 91$ units (125 par). Percentage increase $= \frac{91}{125} \times 100\% = \frac{91 \times 4}{5}\% = 72.8\%$. Atah sahi vikalp (c) 72.8% hai. Method 2 (100-Base Method): Initial volume $= 100$. Pehle do dimensions ka net: $20 + 20 + \frac{20 \times 20}{100} = 44\%$. Ab $44\%$ aur teesri dimension ($20\%$) ka net: $= 44 + 20 + \frac{44 \times 20}{100} = 64 + 8.8 = 72.8\%$. Atah sahi uttar (c) hai.
Suresh owns a rectangular plot. Due to widening of a road near his plot, the length and breadth of his plot are reduced by 20% and 15%, respectively. What is the percentage decrease in the area of his plot?
सुरेश के पास एक आयताकार भूखंड है। उसके भूखंड के पास एक सड़क के चौड़ीकरण के कारण, उसके भूखंड की लंबाई और चौड़ाई में क्रमशः 20% और 15% की कमी हो जाती है। उसके भूखंड के क्षेत्रफल में कितने प्रतिशत की कमी होगी?
(A) (a) 32%
(B) (b) 35%
(C) (c) 30%
(D) (d) 28%
✅ Answer & Explanation
Sahi jawab: A) (a) 32%Explanation: Method 1 (Ratio Method): Length: $20\% = \frac{1}{5} \implies 5 : 4$. Breadth: $15\% = \frac{3}{20} \implies 20 : 17$. Area $=$ Length $\times$ Breadth. Initial Area : New Area $= (5 \times 20) : (4 \times 17) = 100 : 68$. Area 100 se ghatkar 68 ho gaya, isliye kami $= 100 - 68 = 32\%$. Atah sahi vikalp (a) 32% hai. Method 2 (Successive Formula): Net Change $= x + y + \frac{xy}{100}$ Yahan $x = -20\%$ aur $y = -15\%$: $= -20 - 15 + \frac{(-20) \times (-15)}{100} = -35 + 3 = -32\%$ (yaani 32% decrease). Atah sahi uttar (a) hai.
The length and breadth of a cuboid are increased by 10% and 20%, respectively, while its height is reduced by 25%. What is the total percentage increase or decrease in the volume of the cuboid?
एक घनाभ की लंबाई और चौड़ाई में क्रमशः 10% और 20% की वृद्धि की जाती है, जबकि इसकी ऊंचाई में 25% की कमी की जाती है। घनाभ के आयतन में कुल कितने प्रतिशत की वृद्धि या कमी होगी?
(A) (a) Decrease by 1%
(B) (b) Increase by 1%
(C) (c) Decrease by 2%
(D) (d) Increase by 2%
✅ Answer & Explanation
Sahi jawab: A) (a) Decrease by 1%Explanation: Method 1 (Ratio Method): Length ($+10\% = \frac{1}{10}$): ratio $= 10 : 11$. Breadth ($+20\% = \frac{1}{5}$): ratio $= 5 : 6$. Height ($-25\% = -\frac{1}{4}$): ratio $= 4 : 3$. Volume $= l \times b \times h$. Initial Volume : New Volume $= (10 \times 5 \times 4) : (11 \times 6 \times 3) = 200 : 198$. Base 100 banane ke liye 2 se divide karo: $100 : 99$. Volume 100 se 99 ho gaya, yaani $1\%$ ki kami (Decrease by 1%). Atah sahi vikalp (a) Decrease by 1% hai. Method 2 (100-Base Method): Initial Volume $= 100$. Length aur breadth ka net increase: $10 + 20 + \frac{10 \times 20}{100} = 32\%$. Ab $+32\%$ aur height ke $-25\%$ ka net change: $= 32 - 25 + \frac{32 \times (-25)}{100} = 7 - 8 = -1\%$ (1% decrease). Atah sahi uttar (a) hai.
Find the total percentage change in the volume of a cuboid if its length and breadth are decreased by 20% and 10%, respectively, while its height is increased by 50%.
एक घनाभ के आयतन में कुल प्रतिशत परिवर्तन ज्ञात कीजिए यदि इसकी लंबाई और चौड़ाई में क्रमशः 20% और 10% की कमी की जाती है, जबकि इसकी ऊंचाई में 50% की वृद्धि की जाती है।
(A) (a) 8% increase
(B) (b) 8% decrease
(C) (c) 5% decrease
(D) (d) 10% increase
✅ Answer & Explanation
Sahi jawab: A) (a) 8% increaseExplanation: Method 1 (Ratio Method): Length ($-20\% = -\frac{1}{5}$): ratio $= 5 : 4$. Breadth ($-10\% = -\frac{1}{10}$): ratio $= 10 : 9$. Height ($+50\% = +\frac{1}{2}$): ratio $= 2 : 3$. Initial Volume : New Volume $= (5 \times 10 \times 2) : (4 \times 9 \times 3) = 100 : 108$. Volume 100 se badhkar 108 ho gaya, isliye net change $= 108 - 100 = 8\%$ increase. Atah sahi vikalp (a) 8% increase hai. Method 2 (100-Base Method): Initial Volume $= 100$. Length aur breadth dono me kami ka net: $-20 - 10 + \frac{(-20)(-10)}{100} = -30 + 2 = -28\%$. Ab $-28\%$ aur height ke $+50\%$ ka net change: $= -28 + 50 + \frac{(-28) \times 50}{100} = 22 - 14 = +8\%$ (8% increase). Atah sahi uttar (a) hai.
If the length and breadth of a cuboid are increased by 8% and 5%, respectively, while its height remains unchanged, then the percentage increase in its volume is:
यदि एक घनाभ की लंबाई और चौड़ाई में क्रमशः 8% और 5% की वृद्धि की जाती है, जबकि उसकी ऊंचाई अपरिवर्तित रहती है, तो उसके आयतन में कितने प्रतिशत की वृद्धि होगी?
(A) (a) 13.00%
(B) (b) 13.40%
(C) (c) 14.20%
(D) (d) 14.80%
✅ Answer & Explanation
Sahi jawab: B) (b) 13.40%Explanation: Method 1 (Successive Formula): Cuboid ka volume $V = l \times b \times h$. Yahan height constant hai, isliye volume me percentage badlav length aur breadth ke successive change ke barabar hoga: Net Change $= a + b + \frac{ab}{100}$ Yahan $a = +8\%$ aur $b = +5\%$: $= 8 + 5 + \frac{8 \times 5}{100} = 13 + 0.40 = 13.40\%$. Atah sahi vikalp (b) 13.40% hai. Method 2 (Ratio Method): Length ($+8\% = \frac{2}{25}$): ratio $= 25 : 27$. Breadth ($+5\% = \frac{1}{20}$): ratio $= 20 : 21$. Volume ratio $= (25 \times 20) : (27 \times 21) = 500 : 567$. Base ko 100 banane ke liye 5 se divide karo: $100 : 113.4$. Volume me increase $= 113.4 - 100 = 13.4\%$. Atah sahi uttar (b) hai.
If the height of a cylinder is decreased by 20% and its radius is increased by 20%, then the volume changes by:
यदि एक बेलन की ऊंचाई 20% कम कर दी जाए और त्रिज्या 20% बढ़ा दी जाए, तो आयतन में कितना परिवर्तन होगा?
(A) (a) Increase by 15.2%
(B) (b) Decrease by 15.2%
(C) (c) Increase by 12.5%
(D) (d) No change
✅ Answer & Explanation
Sahi jawab: A) (a) Increase by 15.2%Explanation: Method 1 (Ratio Method): Cylinder ka volume $V = \pi r^2 h \implies V \propto r \times r \times h$. Radius ($+20\% = +\frac{1}{5}$): ratio $= 5 : 6$ (do baar apply hoga). Height ($-20\% = -\frac{1}{5}$): ratio $= 5 : 4$. Initial Volume : New Volume $= (5 \times 5 \times 5) : (6 \times 6 \times 4) = 125 : 144$. Volume me increase $= 144 - 125 = 19$ units (125 ke base par). Percentage change $= \frac{19}{125} \times 100\% = \frac{19 \times 4}{5}\% = 15.2\%$ increase. Atah sahi vikalp (a) Increase by 15.2% hai. Method 2 (Successive Formula): Radius me do baar $20\%$ badhav ka net: $20 + 20 + \frac{20 \times 20}{100} = 44\%$. Ab $+44\%$ aur height ke $-20\%$ ka net change: $= 44 - 20 + \frac{44 \times (-20)}{100} = 24 - 8.8 = +15.2\%$ (Increase by 15.2%). Atah sahi uttar (a) hai.
The radius of a sphere is reduced by 20%. By what approximate percentage does the volume decrease?
एक गोले की त्रिज्या 20% कम कर दी जाती है। आयतन में लगभग कितने प्रतिशत की कमी होगी?
(A) (a) 48.8%
(B) (b) 51.2%
(C) (c) 45.5%
(D) (d) 52.8%
✅ Answer & Explanation
Sahi jawab: A) (a) 48.8%Explanation: Method 1 (Ratio Method): Sphere ka volume $V = \frac{4}{3}\pi r^3 \implies V \propto r^3$. Radius reduction $= 20\% = \frac{1}{5}$. Initial Radius : New Radius $= 5 : (5 - 1) = 5 : 4$. Volume ratio $= 5^3 : 4^3 = 125 : 64$. Volume me kami $= 125 - 64 = 61$ units (125 ke base par). Percentage decrease $= \frac{61}{125} \times 100\% = \frac{61 \times 4}{5}\% = \frac{244}{5}\% = 48.8\%$. Atah sahi vikalp (a) 48.8% hai. Method 2 (100-Base Method): Initial volume $= 100$. Pehle do radius cut ka net: $-20 - 20 + \frac{(-20)(-20)}{100} = -40 + 4 = -36\%$. Ab $-36\%$ aur teesre cut ($-20\%$) ka net: $= -36 - 20 + \frac{(-36) \times (-20)}{100} = -56 + 7.2 = -48.8\%$ (yaani 48.8% decrease). Atah sahi uttar (a) hai.
If both the length and breadth of a cuboid are increased by 20%, then by how much percent (correct to two decimal places) should its height be reduced so that the volume of the cuboid remains the same?
यदि एक घनाभ की लंबाई और चौड़ाई दोनों में 20% की वृद्धि की जाती है, तो इसकी ऊंचाई में कितने प्रतिशत की कमी की जानी चाहिए ताकि घनाभ का आयतन समान रहे?
(A) (a) 28.55%
(B) (b) 30.56%
(C) (c) 32.25%
(D) (d) 33.33%
✅ Answer & Explanation
Sahi jawab: B) (b) 30.56%Explanation: Method 1 (Ratio Method): $\text{Volume} = \text{Base Area} \times \text{Height} = \text{Constant} \implies \text{Height} \propto \frac{1}{\text{Base Area}}$. Length aur breadth dono $20\% (\frac{1}{5})$ badhe hain: Base Area ratio $= (5 \times 5) : (6 \times 6) = 25 : 36$. Volume unchanged rakhne ke liye height ka ratio ulta hoga: Height ratio $= 36 : 25$. Height me reduction $= 36 - 25 = 11$ units (36 ke base par). Percentage reduction $= \frac{11}{36} \times 100\% = \frac{275}{9}\% \approx 30.56\%$. Atah sahi vikalp (b) 30.56% hai. Method 2 (100-Base Method): Initial base area $= 100$. Length aur breadth badhne ke baad naya base area: $= 20 + 20 + \frac{20 \times 20}{100} = 44\% \implies 144$. Volume constant rakhne ke liye height ko 144 se ghata kar wapas 100 par laana hoga: Reduction needed $= 144 - 100 = 44$. 144 ke base par percentage reduction: $= \frac{44}{144} \times 100\% = \frac{11}{36} \times 100\% \approx 30.56\%$. Atah sahi uttar (b) hai.