Questions and Answers
If the price of a book is increased by 20% and then decreased by 10%, what is the net percentage change in the price?
8% decrease
A student scored 65% marks in one subject and 75% marks in another subject. What is the overall percentage of marks obtained by the student if both subjects carry equal weightage?
70%
If the final salary is $10,500 after a 10% increase in the first year and a 5% decrease in the second year, what was the initial salary?
$10,000
If the price of a commodity increases by 25%, by what percent must its consumption be reduced to not increase the expenditure?
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If 60% of 240 students passed an examination, how many students failed?
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If a shopkeeper sold a T-shirt at a 25% profit with a cost price of $40, what was the selling price?
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If an amount doubles in 5 years at simple interest, what is the rate of interest per annum?
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Find the smallest positive integer n such that $n^2 + 25$ is divisible by 5.
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If a + b + c = 6, ab + bc + ca = 11, and abc = 6, find the value of $a^3 + b^3 + c^3 - 3abc$.
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In triangle ABC where angle A = 90°, AB = 6, and BC = 8, what is the length of AC?
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In how many ways can 5 boys and 5 girls be seated in a row such that no two boys are together?
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From a box with 5 red balls, 4 blue balls, and 3 green balls, if 2 balls are drawn randomly, what is the probability that both are blue?
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If a cone is inscribed in a hemisphere with a hemisphere radius of 6 cm, what is the volume of the cone?
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A can do a piece of work in 6 days, B can do it in 8 days. They work together for 3 days and then A leaves. In how many more days will B finish the remaining work?
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A shopkeeper marks his goods 50% above cost price, then allows a 20% discount and sells for $160. Find the cost price of the article.
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If a city's population increased by 20% in the first year and decreased by 10% in the second year, what is the net percentage increase or decrease in population?
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A cuboid with dimensions 6 cm x 8 cm x 10 cm has a cube cut from it. What is the surface area of the remaining part?
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A solid sphere of radius 6 cm is melted and recast into small spheres of radius 3 cm. How many small spheres can be formed?
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A cylindrical pipe with inner radius 4 cm and thickness 1 cm, and length 50 cm. What is the volume of the material used in making the pipe?
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A conical vessel with height 16 cm and base diameter 14 cm is full of water. Find the volume of the water in the vessel.
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A frustum of a cone with diameters 12 cm and 20 cm and height 16 cm. Find its volume.
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A hemisphere is cut out from a solid wooden cube with side 21 cm. What is the surface area of the remaining solid?
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A solid wooden cube of side 10 cm is cut into eight cubes of equal volume. What is the total surface area of all the small cubes?
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A regular square pyramid with base side length 6 cm and slant height 8 cm. Find its total surface area.
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A cylindrical glass filled with water to the brim has diameter 14 cm and height 10 cm. If the water is poured into a rectangular vessel with base area 140 sq.cm, find the height of the water level in the vessel.
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An irregular block of wood 10 cm long, 6 cm wide, 4 cm high is cut into small cubes of side 2 cm. What is the total number of cubes?
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A sum of $5000 is lent at a certain rate of interest for 2 years, compounded annually. If the amount after 2 years is $5500, find the rate of interest.
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Find the compound interest on $10,000 at 10% per annum for 2 years, compounded quarterly.
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John invested a sum of money at 5% interest compounded annually. After 4 years, he received $3000. Find the sum he invested.
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A sum of money amounts to $9680 in 2 years and $10648 in 3 years when compounded annually. Find the sum and the rate of interest.
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The simple interest on a certain sum for 3 years at 12% per annum is $5400. Find the sum.
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Find the compound interest on $5000 at 10% per annum for 2 years, compounded half-yearly.
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A sum of money becomes 3 times itself in 5 years at simple interest. What is the annual rate of interest?
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Find the compound interest on $4000 at 8% per annum for 2 years, compounded annually.
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A sum of money becomes 5/4 times itself in 2 years at simple interest. What is the rate of interest?
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At what rate of interest per annum will a sum of money double itself in 5 years, compounded annually?
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In triangle ABC where angle A = 60°, angle B = 75°, and AC = 6 cm, find the length of side BC.
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In a rectangle, if the length is 6 cm more than the breadth and the diagonal is 10 cm, find the area.
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In a cyclic quadrilateral ABCD where AB = 6 cm, BC = 8 cm, CD = 4 cm, and angle ADC = 90°, find the length of AD.
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Given the radius of the incircle of a triangle is 4 cm and the radius of the circumcircle is 5 cm, find the area of the triangle.
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In a regular hexagon, find the ratio of the area of the inscribed circle to the area of the circumscribed circle.
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A triangle has sides of lengths 13 cm, 14 cm, and 15 cm. Find the length of the altitude drawn to the side of length 14 cm.
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Two tangents to a circle from an external point are of lengths 8 cm and 12 cm. If the distance between the centers of the circle and the point is 10 cm, find the radius of the circle.
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A parallelogram has adjacent sides of lengths 10 cm and 12 cm. If the distance between the longer sides is 8 cm, find the area.
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In a trapezium ABCD where AB || DC, AB = 8 cm, DC = 6 cm, and the height is 5 cm, find the length of side BC.
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A square and a circle have the same perimeter. Find the ratio of the area of the square to the area of the circle.
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Study Notes
Here are the study notes for the provided text:
- Percentages*
- If the price of a book is increased by 20% and then decreased by 10%, the net percentage change in the price is an 8% increase.
- A student scored 65% marks in one subject and 75% marks in another subject, and the overall percentage of marks obtained is the average of the two subjects.
- If a salary is increased by 10% in the first year and decreased by 5% in the second year, the final salary can be found by applying the percentage changes successively.
- If the price of a commodity increases by 25%, the consumption must be reduced by 20% to not increase the expenditure.
- In an election, if 60% of the students passed, the number of students that failed can be found by subtracting the number of students that passed from the total number of students.
- Permutations and Combinations*
- A committee of 4 members can be chosen from 10 men and 5 women if the committee must consist of at least 2 men and 1 woman.
- Different words can be formed using the letters of the word "UNIQUE".
- A student can select 4 books if at least 2 books must be mathematics books.
- Different arrangements can be made from the letters of the word "COMPUTER" if the vowels must occupy the odd positions.
- A president, vice president, and treasurer can be selected from a group of 8 people if one person cannot hold more than one position.
- Different passwords can be formed using the English alphabet and digits 0-9 if repetition of characters is allowed.
- Clocks*
- The hands of a clock are together at 5:27:16 2/11 minutes past 5.
- The hands of a clock form a right angle 44 times in a day.
- The acute angle between the hour and minute hands of a clock at 4:20 is 120°.
- The hands of a clock overlap 22 times in a day.
- The minute hand and hour hand of a clock are perpendicular to each other at 7:21:49 1/11 minutes past 7.
- Calendars*
- 25 days ago, if today is Tuesday, the day of the week was Thursday.
- If 8th March 2024 is a Friday, 8th March 2044 is a Monday.
- The 13th of the month occurs 2 times in a non-leap year.
- If January 1, 2025, falls on a Tuesday, October 1, 2025, is a Wednesday.
- 333 days from today, if today is Monday, the day of the week is Wednesday.
- Time and Work*
- A, B, and C together can complete a piece of work in 4 days.
- A, B, and C together can complete a piece of work in 6 days.
- A and B together can complete a piece of work in 7.5 days.
- B and C together can complete a piece of work in 15 days.
- B can complete a piece of work in 12 days.
- Distance and Speed*
- The average speed of a car for the whole journey is 48 km/h.
- A train covers a distance of 600 km in 5 hours.
- A cyclist travels from town A to town B at a speed of 15 km/h.
- A car travels from city X to city Y at an average speed of 50 km/h.
- A boat travels upstream for 3 hours and covers a distance of 30 km.
- Mixed Questions*
- The smallest positive integer n such that n^2 + 25 is divisible by 5 is 5.
- a + b + c = 6, ab + bc + ca = 11, and abc = 6, then a^3 + b^3 + c^3 - 3abc = 72.
- In triangle ABC, angle A = 90°, AB = 6, and BC = 8, then the length of AC is 10.
- 5 boys and 5 girls can be seated in a row such that no two boys are together in 2400 ways.
- The probability that both balls are blue is 2/65.
- The volume of the cone is 144π / 5 cubic cm.
- Surface Area and Volumes*
- The surface area of the remaining part is 424 sq.cm.
- 64 small spheres can be formed from a solid sphere of radius 6 cm.
- The volume of the material used in making the pipe is 692Π cm^3.
- The volume of water in the vessel is 1232π cm³.
- The volume of the frustum of the cone is 400π/3 cm^3.
- The surface area of the remaining solid is 441π sq.cm.
- Simple and Compound Interest*
- The rate of interest is 10% per annum.
- The compound interest is $2104.16.
- The sum invested is $2500.
- The sum and the rate of interest are $10,000 at 4% per annum.
- The simple interest is $5400.
- Geometry*
- The length of side BC is 2 root7 cm.
- The area of the rectangle is 24 sq.cm.
- The length of AD is 4 cm.
- The area of the triangle is 24 sq.cm.
- The ratio of the area of the inscribed circle to the area of the circumscribed circle is Root3 /2 : root3 /3.
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