CBSE Class 6 Mathematics Revision Worksheet Set 07

Question 1. What fraction is 1 hour of 1 day?
Answer: Since 1 day = 24 hours, 1 hour of 1 day = \(\frac{1}{24}\)
In simple words: One hour is one part out of 24 equal parts that make up a whole day.

Exam Tip: Always identify the total units before writing the fraction.

Question 2. What fraction is 50 paise of Rs. 8?
Answer: Since Rs. 8 = 800 paise, 50 paise of Rs. 8 = \(\frac{50}{800} = \frac{1}{16}\)
In simple words: Convert both quantities to the same unit first, then express as a fraction and simplify.

Exam Tip: Remember: 1 Rupee = 100 paise. Always convert to the same unit before forming fractions.

Question 3. Represent \(\frac{5}{4}\) on the number line.
Answer: \(\frac{5}{4} = 1\frac{1}{4} = 1.25\). On a number line, mark points 0, 1, 2. Divide the space between 1 and 2 into 4 equal parts. The point \(\frac{5}{4}\) lies at 1 full unit plus 1 division into the next unit.
In simple words: Convert the improper fraction to a mixed number, then locate it between two whole numbers on the line.

Exam Tip: For improper fractions, first convert to mixed numbers to easily identify the position on the number line.

Question 4a. Express as improper fraction: \(5\frac{1}{11}\)
Answer: \(5\frac{1}{11} = \frac{(5 \times 11) + 1}{11} = \frac{55 + 1}{11} = \frac{56}{11}\)
In simple words: Multiply the whole number by the denominator, add the numerator, and keep the same denominator.

Exam Tip: Formula: Mixed fraction \(a\frac{b}{c} = \frac{(a \times c) + b}{c}\)

Question 4b. Express as improper fraction: \(7\frac{2}{7}\)
Answer: \(7\frac{2}{7} = \frac{(7 \times 7) + 2}{7} = \frac{49 + 2}{7} = \frac{51}{7}\)
In simple words: Multiply the whole number by the denominator, add the numerator, and keep the same denominator.

Exam Tip: The improper fraction will always have a numerator larger than the denominator.

Question 5a. Express as mixed fraction: \(\frac{21}{5}\)
Answer: \(\frac{21}{5} = 4\frac{1}{5}\) (Since 21 ÷ 5 = 4 remainder 1)
In simple words: Divide the numerator by the denominator; the quotient becomes the whole number and the remainder becomes the new numerator.

Exam Tip: Use long division: numerator ÷ denominator = whole number R remainder, then write as whole \(\frac{\text{remainder}}{\text{denominator}}\)

Question 5b. Express as mixed fraction: \(\frac{40}{9}\)
Answer: \(\frac{40}{9} = 4\frac{4}{9}\) (Since 40 ÷ 9 = 4 remainder 4)
In simple words: Divide the numerator by the denominator; the quotient becomes the whole number and the remainder becomes the new numerator.

Exam Tip: Always check your answer by converting back: \(4\frac{4}{9} = \frac{(4 \times 9) + 4}{9} = \frac{40}{9}\)

Question 6a. Write 4 equivalent fractions for \(\frac{3}{5}\)
Answer: \(\frac{3}{5} = \frac{6}{10} = \frac{9}{15} = \frac{12}{20} = \frac{15}{25}\)
In simple words: Multiply both numerator and denominator by the same number to get equivalent fractions.

Exam Tip: To create equivalent fractions, always multiply (or divide) both numerator and denominator by the same non-zero number.

Question 6b. Write 4 equivalent fractions for \(\frac{9}{11}\)
Answer: \(\frac{9}{11} = \frac{18}{22} = \frac{27}{33} = \frac{36}{44} = \frac{45}{55}\)
In simple words: Multiply both numerator and denominator by the same number to get equivalent fractions.

Exam Tip: You can multiply by 2, 3, 4, 5, etc. to get different equivalent fractions.

Question 7a. Fill up: \(\frac{18}{25} = \frac{a}{75}\)
Answer: Cross multiply: \(18 \times 75 = 25 \times a\) → \(1350 = 25a\) → \(a = 54\)
Therefore: \(\frac{18}{25} = \frac{54}{75}\)
In simple words: Find what number the denominator is multiplied by, then multiply the numerator by the same number.

Exam Tip: Check: 75 ÷ 25 = 3, so multiply numerator by 3: 18 × 3 = 54

Question 7b. Fill up: \(\frac{21}{77} = \frac{3}{y}\)
Answer: Cross multiply: \(21 \times y = 77 \times 3\) → \(21y = 231\) → \(y = 11\)
Therefore: \(\frac{21}{77} = \frac{3}{11}\)
In simple words: Use cross multiplication to find the unknown denominator.

Exam Tip: Simplify by finding the common factor: 21 ÷ 7 = 3 and 77 ÷ 7 = 11

Question 8a. Compare: \(\frac{6}{7}\) and \(\frac{5}{8}\)
Answer: Find common denominator (56): \(\frac{6}{7} = \frac{48}{56}\) and \(\frac{5}{8} = \frac{35}{56}\)
Since 48 > 35, \(\frac{6}{7} > \frac{5}{8}\)
In simple words: Convert both fractions to have the same denominator, then compare the numerators.

Exam Tip: The common denominator is the LCM of the two denominators. Here: LCM(7, 8) = 56

Question 8b. Compare: \(\frac{8}{19}\) and \(\frac{2}{9}\)
Answer: Cross multiply: \(8 \times 9 = 72\) and \(2 \times 19 = 38\)
Since 72 > 38, \(\frac{8}{19} > \frac{2}{9}\)
In simple words: Cross multiply both fractions and compare the products.

Exam Tip: Quick method: If \(\frac{a}{b}\) and \(\frac{c}{d}\), then \(a \times d\) vs \(b \times c\)

Question 9a. Find: \(1\frac{2}{3} + 2\frac{3}{4}\)
Answer: Convert to improper fractions: \(\frac{5}{3} + \frac{11}{4}\)
Find common denominator (12): \(\frac{20}{12} + \frac{33}{12} = \frac{53}{12} = 4\frac{5}{12}\)
In simple words: Convert mixed numbers to improper fractions, find a common denominator, add, and convert back to a mixed number.

Exam Tip: The common denominator of 3 and 4 is 12 (their LCM).

Question 9b. Find: \(7\frac{4}{9} - 3\frac{5}{18}\)
Answer: Convert to improper fractions: \(\frac{67}{9} - \frac{59}{18}\)
Find common denominator (18): \(\frac{134}{18} - \frac{59}{18} = \frac{75}{18} = \frac{25}{6} = 4\frac{1}{6}\)
In simple words: Convert mixed numbers to improper fractions, find a common denominator, subtract, and simplify.

Exam Tip: Always simplify the final answer. Here: \(\frac{75}{18} = \frac{25}{6}\) by dividing by 3.

Question 1 (Decimals). Convert into fractions in lowest form: 0.75
Answer: \(0.75 = \frac{75}{100} = \frac{3}{4}\) (Divide both by 25)
In simple words: Write the decimal as a fraction with the appropriate power of 10 in the denominator, then simplify.

Exam Tip: Two decimal places means denominator is 100. Find the GCD to simplify.

Question 1b (Decimals). Convert into fractions in lowest form: 30.04
Answer: \(30.04 = \frac{3004}{100} = \frac{751}{25}\) (Divide both by 4)
In simple words: The decimal 30.04 has two decimal places, so write with denominator 100, then simplify.

Exam Tip: To simplify, find the GCD. Here: GCD(3004, 100) = 4

Question 2a (Decimals). Convert into decimal: \(\frac{4}{5}\)
Answer: \(\frac{4}{5} = \frac{8}{10} = 0.8\)
In simple words: Multiply numerator and denominator to make the denominator a power of 10, then write as a decimal.

Exam Tip: To convert fractions to decimals, multiply to get 10, 100, or 1000 in the denominator.

Question 2b (Decimals). Convert into decimal: \(\frac{75}{1000}\)
Answer: \(\frac{75}{1000} = 0.075\)
In simple words: Denominator 1000 means three decimal places. The numerator 75 becomes 0.075.

Exam Tip: Denominator of 1000 = 3 decimal places, 100 = 2 places, 10 = 1 place.

Question 2c (Decimals). Convert into decimal: \(\frac{12}{100}\)
Answer: \(\frac{12}{100} = 0.12\)
In simple words: Denominator 100 means two decimal places. The numerator 12 becomes 0.12.

Exam Tip: Count decimal places from right to left based on the denominator.

Question 3a (Decimals). Write the place value of 9 in 395.017
Answer: The digit 9 is in the tens place. Place value of 9 = \(9 \times 10 = 90\)
In simple words: Identify the position of the digit and multiply by its place value.

Exam Tip: Place values from left: hundreds (100), tens (10), ones (1), tenths (0.1), hundredths (0.01), thousandths (0.001)

Question 3b (Decimals). Write the place value of 9 in 213.095
Answer: The digit 9 is in the hundredths place. Place value of 9 = \(9 \times 0.01 = 0.09\)
In simple words: Identify the position of the digit and multiply by its place value.

Exam Tip: After the decimal point, the first digit is tenths, second is hundredths, third is thousandths.

Question 3c (Decimals). Write the place value of 9 in 103.095
Answer: The digit 9 is in the hundredths place. Place value of 9 = \(9 \times 0.01 = 0.09\)
In simple words: The digit 9 occupies the hundredths position in the decimal part.

Exam Tip: Always count positions carefully after the decimal point.

Question 4a (Decimals). Fill using < , > , = : 3.52 __ 3.521
Answer: \(3.52 < 3.521\) (Or 3.520 < 3.521)
In simple words: Compare decimal numbers by comparing each place value from left to right.

Exam Tip: Add zeros to make decimals have the same number of places: 3.520 vs 3.521

Question 4b (Decimals). Fill using < , > , = : 100.1 __ 100.02
Answer: \(100.1 > 100.02\) (Or 100.10 > 100.02)
In simple words: Compare the tenths place: 1 tenth is greater than 0 tenths.

Exam Tip: Compare place by place: 100.10 has 1 in tenths, 100.02 has 0 in tenths.

Question 4c (Decimals). Fill using < , > , = : 12.9 __ 12.90
Answer: \(12.9 = 12.90\)
In simple words: Adding zeros to the right of a decimal does not change its value.

Exam Tip: 12.9 is the same as 12.90, which is the same as 12.900.

Question 5a (Decimals). Convert into like decimals: 4.3, 2.85, 3.001, 7.05, 8.9, 9.21
Answer: The maximum decimal places is 3 (in 3.001). Like decimals: \(4.300, 2.850, 3.001, 7.050, 8.900, 9.210\)
In simple words: Add zeros to the right so all decimals have the same number of decimal places.

Exam Tip: Count decimal places in each number and find the maximum, then add zeros to others.

Question 5b (Decimals). Convert into like decimals: 2.111, 4.0131, 1.08, 3.007, 7.1
Answer: The maximum decimal places is 4 (in 4.0131). Like decimals: \(2.1110, 4.0131, 1.0800, 3.0070, 7.1000\)
In simple words: Add zeros to match the number with the most decimal places.

Exam Tip: This helps when adding or subtracting decimals with different lengths.

Question 6a (Decimals). Find the sum: 0.6 + 0.5
Answer: \(0.6 + 0.5 = 1.1\)
In simple words: Add the decimal numbers place by place as you would with whole numbers.

Exam Tip: Align the decimal points vertically before adding.

Question 6b (Decimals). Find the sum: 8.9 + 3.58 + 7.213
Answer: Convert to like decimals: \(8.900 + 3.580 + 7.213 = 19.693\)
In simple words: Make all decimals have the same number of places, then add them vertically.

Exam Tip: Always align decimal points and add as if adding whole numbers, then place the decimal in the result.

Question 7a (Decimals). Subtract 0.035 from 0.098
Answer: \(0.098 - 0.035 = 0.063\)
In simple words: Subtract the smaller decimal from the larger one by aligning decimal points.

Exam Tip: Always put the larger number on top when subtracting.

Question 7b (Decimals). Subtract 45.07 from 73.1
Answer: Convert to like decimals: \(73.10 - 45.07 = 28.03\)
In simple words: Make both decimals have the same number of places, then subtract by aligning decimal points.

Exam Tip: Rewrite as: 73.10 - 45.07, then subtract place by place.

Question 8 (Decimals). The height of a mango tree is 3.13 m and orange tree is 1.87 m. Which is higher and by how much?
Answer: The mango tree is higher. Difference = \(3.13 - 1.87 = 1.26\) m. The mango tree is 1.26 m higher than the orange tree.
In simple words: Compare the two heights and subtract the smaller from the larger to find the difference.

Exam Tip: Always subtract the smaller value from the larger value to get a positive result.

Question 1 (Data Handling). Heights of 30 students (in cm): 143, 141, 142, 143, 145, 142, 144, 141, 140, 142, 141, 140, 143, 139, 144, 145, 142, 140, 141, 143, 142, 144, 145, 142, 140, 142, 141, 144, 143, 140. Prepare a tally marks table and find: a) students with maximum height, b) students with minimum height
Answer: Tally marks table:
Height (cm) | Tally Marks | Frequency
139 | | | 1
140 | | | | | 4
141 | | | | | 4
142 | | | | | | | 6
143 | | | | | 4
144 | | | | | 4
145 | | | 3

a) Maximum height (145 cm): 3 students
b) Minimum height (139 cm): 1 student
In simple words: Count how many times each height appears and mark with tally marks, then identify the highest and lowest frequencies.

Exam Tip: Use tally marks (groups of 5 with one diagonal line) to count efficiently. Every 5th mark is drawn diagonally across 4 vertical marks.

Question 2 (Data Handling). Bar graph for trees planted by classes: Class 6-75, Class 7-60, Class 8-105, Class 9-90, Class 10-45
Answer: To draw a bar graph:
1. Draw two perpendicular axes (X and Y)
2. Mark classes (6, 7, 8, 9, 10) on X-axis
3. Mark "No. of trees" from 0 to 110 on Y-axis
4. Draw vertical bars: Class 6 (75), Class 7 (60), Class 8 (105), Class 9 (90), Class 10 (45)
5. Use equal width bars with space between them
In simple words: Create vertical bars representing the number of trees for each class, with height proportional to the values.

Exam Tip: Use a scale (e.g., 1 cm = 10 trees) to ensure accuracy. Keep equal spacing between bars and same width for all bars.

Question 3 (Data Handling). Ages of 25 students: 15, 13, 16, 12, 14, 15, 16, 13, 12, 14, 16, 16, 15, 12, 11, 15, 16, 13, 12, 14, 15, 15, 12, 11, 16. Prepare tally table and draw pictograph (1 symbol = 2 students)
Answer: Tally marks table:
Age | Tally Marks | Frequency
11 | | | 2
12 | | | | | | | 5
13 | | | 3
14 | | | 3
15 | | | | | | | 6
16 | | | | | | | 6

Pictograph (using 1 symbol ☺ = 2 students):
Age 11: ☺ (2 students)
Age 12: ☺ ☺ ☺ (5 students, so 2.5 symbols - use half symbol)
Age 13: ☺ ☺ (3 students, so 1.5 symbols)
Age 14: ☺ ☺ (3 students, so 1.5 symbols)
Age 15: ☺ ☺ ☺ (6 students)
Age 16: ☺ ☺ ☺ (6 students)
In simple words: Count frequencies using tally marks, then represent with symbols where each symbol represents 2 students.

Exam Tip: Use consistent symbols throughout and clearly indicate what each symbol represents in a key.

Question 1 (Mensuration). Perimeter of rectangle = 84 m, breadth = 17 m. Find length.
Answer: Using Perimeter = \(2(l + b)\)
\(84 = 2(l + 17)\)
\(42 = l + 17\)
\(l = 25\) m
In simple words: Use the perimeter formula to set up an equation, then solve for the unknown length.

Exam Tip: Perimeter of rectangle = 2(length + breadth). Always substitute known values and solve step by step.

Question 2 (Mensuration). Perimeter of square = 50 m. Find the side.
Answer: Using Perimeter = \(4s\) (where s = side)
\(50 = 4s\)
\(s = \frac{50}{4} = 12.5\) m
In simple words: A square has 4 equal sides, so divide the perimeter by 4 to find one side.

Exam Tip: Perimeter of square = 4 × side. Remember all sides of a square are equal.

Question 3 (Mensuration). Side of square = 20 m. Find cost of fencing at Rs 15 per metre.
Answer: Perimeter = \(4 \times 20 = 80\) m
Cost = \(80 \times 15 = Rs\) 1200
In simple words: Calculate the perimeter to find the total length to be fenced, then multiply by cost per metre.

Exam Tip: Fencing covers the perimeter, not the area. Always use perimeter for boundary-related problems.

Question 4 (Mensuration). Room 12 m long and 8 m wide. Find tiles needed (square tile of side 50 cm each).
Answer: Room area = \(12 \times 8 = 96\) m² = \(96 \times 10000 = 960000\) cm²
Tile area = \(50 \times 50 = 2500\) cm²
Number of tiles = \(\frac{960000}{2500} = 384\) tiles
In simple words: Calculate the room area and tile area in the same units, then divide room area by tile area.

Exam Tip: Convert all measurements to the same unit before calculating. 1 m = 100 cm, so 1 m² = 10000 cm².

Question 5a (Mensuration). Tick correct option: The total length around a closed figure is called:
Answer: (b) Perimeter
In simple words: Perimeter is the total distance around the boundary of any closed figure.

Exam Tip: Remember: Perimeter = boundary length, Area = space inside, Circumference = perimeter of a circle.

Question 5b (Mensuration). Tick correct option: Area of rectangle is
Answer: (a) l × b
In simple words: Area of rectangle = length × breadth. Option (b) is for area of square, and (c) is for perimeter.

Exam Tip: For rectangle: Area = l × b, Perimeter = 2(l + b). For square: Area = s², Perimeter = 4s.

Question 5c (Mensuration). Tick correct option: Perimeter of regular pentagon of side 3 cm is
Answer: (b) 15 cm
Explanation: A pentagon has 5 sides. Perimeter = \(5 \times 3 = 15\) cm
In simple words: For a regular polygon, perimeter = number of sides × length of one side.

Exam Tip: Pentagon = 5 sides, Hexagon = 6 sides, Heptagon = 7 sides. For regular polygons, all sides are equal.

Question 1a (Algebra). Write as mathematical expression: The sum of 11 and x
Answer: \(11 + x\)
In simple words: "Sum" means addition. The sum of 11 and x = 11 + x

Exam Tip: Key words: sum (+), difference (-), product (×), quotient (÷)

Question 1b (Algebra). Write as mathematical expression: A number x increased by 8
Answer: \(x + 8\)
In simple words: "Increased by" means addition. A number x increased by 8 = x + 8

Exam Tip: "Increased by" means +, "decreased by" means -, "times" means ×, "divided by" means ÷

Question 1c (Algebra). Write as mathematical expression: One hundred divided by a number p
Answer: \(\frac{100}{p}\) or \(100 \div p\)
In simple words: "Divided by" means division. One hundred divided by p = 100 ÷ p

Exam Tip: Division can be written as a fraction or using the ÷ symbol.

Question 1d (Algebra). Write as mathematical expression: 5 less than twice a number y
Answer: \(2y - 5\)
In simple words: "Twice" means 2×, "less than" means subtract. 5 less than twice y = 2y - 5

Exam Tip: Be careful with order: "5 less than 2y" means subtract 5 from 2y, not the other way around.

Question 1e (Algebra). Write as mathematical expression: One subtracted from five times a number multiplied by two
Answer: \(2(5x - 1)\) or \(10x - 2\)
Explanation: Five times a number = 5x, One subtracted from it = 5x - 1, multiplied by two = 2(5x - 1) = 10x - 2
In simple words: Work through the phrase step by step: 5x, then subtract 1, then multiply the result by 2.

Exam Tip: Follow order of operations and carefully interpret the phrasing from left to right.

Question 2 (Algebra). Find a rule for cost of x books if one book costs Rs 29
Answer: Cost of x books = \(29x\)
In simple words: If one book costs 29 rupees, then x books cost 29 times x, which is 29x rupees.

Exam Tip: For proportional relationships, multiply the rate by the quantity. Here: rate (29) × quantity (x) = 29x

Question 3 (Algebra). Find a rule for perimeter of regular hexagon with each side y cm
Answer: Perimeter = \(6y\) cm
Explanation: A hexagon has 6 sides. If each side = y cm, then perimeter = 6 × y = 6y cm
In simple words: A regular hexagon has 6 equal sides. Multiply the side length by 6.

Exam Tip: Hexagon = 6 sides. For any regular polygon, perimeter = number of sides × side length.

Question 4a (Algebra). Find the value: 3x - 4 when x = 2
Answer: Substitute x = 2: \(3(2) - 4 = 6 - 4 = 2\)
In simple words: Replace x with 2 in the expression, then calculate following order of operations.

Exam Tip: Always substitute the value in place of the variable and calculate carefully.

Question 4b (Algebra). Find the value: 3x - 4 when x = 7
Answer: Substitute x = 7: \(3(7) - 4 = 21 - 4 = 17\)
In simple words: Replace x with 7 in the expression, then calculate.

Exam Tip: Follow order of operations: multiplication before subtraction.

Question 4c (Algebra). Find the value: 3x - y + 2z when x = 2, y = 3, z = 4
Answer: Substitute x = 2, y = 3, z = 4:
\(3(2) - 3 + 2(4) = 6 - 3 + 8 = 11\)
In simple words: Replace each variable with its value, then calculate step by step following order of operations.

Exam Tip: With multiple variables, substitute all values first, then calculate left to right for addition/subtraction.

Question 1a (Ratio and Proportion). Annual income = Rs 250000, savings = Rs 50000. Find ratio of savings to total income
Answer: Ratio = \(\frac{50000}{250000} = \frac{1}{5} = 1:5\)
In simple words: Divide savings by income to get the ratio, then simplify to lowest terms.

Exam Tip: Always simplify ratios to their simplest form by dividing by the GCD.

Question 1b (Ratio and Proportion). Annual income = Rs 250000, savings = Rs 50000. Find ratio of savings to expenditure
Answer: Expenditure = \(250000 - 50000 = 200000\)
Ratio = \(\frac{50000}{200000} = \frac{1}{4} = 1:4\)
In simple words: First find expenditure by subtracting savings from income, then find the ratio.

Exam Tip: Income = Savings + Expenditure. Always identify what you need to find before setting up the ratio.

Question 1c (Ratio and Proportion). Annual income = Rs 250000, savings = Rs 50000. Find ratio of income to expenditure
Answer: Expenditure = \(250000 - 50000 = 200000\)
Ratio = \(\frac{250000}{200000} = \frac{5}{4} = 5:4\)
In simple words: Find expenditure first, then express the ratio of income to expenditure in simplest form.

Exam Tip: The ratio 250000:200000 simplifies to 5:4 by dividing both by 50000.

Question 2a (Ratio and Proportion). Find x: 4:16::x:8
Answer: Using property: \(\frac{4}{16} = \frac{x}{8}\)
Cross multiply: \(4 \times 8 = 16 \times x\)
\(32 = 16x\)
\(x = 2\)
In simple words: In a proportion, cross multiply and solve for the unknown.

Exam Tip: A:B::C:D means A/B = C/D. Cross multiply: A×D = B×C

Question 2b (Ratio and Proportion). Find x: x:6::24:32
Answer: Using property: \(\frac{x}{6} = \frac{24}{32}\)
Cross multiply: \(x \times 32 = 6 \times 24\)
\(32x = 144\)
\(x = 4.5\)
In simple words: Cross multiply and divide to find the unknown value.

Exam Tip: Check: 4.5:6 = 0.75 and 24:32 = 0.75. Both ratios are equal.

Question 3 (Ratio and Proportion). A dozen erasers cost Rs 54. How many erasers for Rs 99?
Answer: Cost per eraser = \(\frac{54}{12} = 4.5\) Rs per eraser
Number of erasers for Rs 99 = \(\frac{99}{4.5} = 22\) erasers
In simple words: Find the cost of one eraser, then divide total money by cost per eraser.

Exam Tip: Always find the unit rate (cost/quantity for one item) first, then use it to solve.

Question 4 (Ratio and Proportion). Train covers 300 km in 4 hours. Find distance in 9 hours.
Answer: Speed = \(\frac{300}{4} = 75\) km/hour
Distance in 9 hours = \(75 \times 9 = 675\) km
In simple words: Find the speed by dividing distance by time, then multiply speed by the new time.

Exam Tip: For uniform speed problems: Speed = Distance/Time, Distance = Speed × Time

Question 5 (Ratio and Proportion). Divide Rs 5850 between Suvidhea and Shweta in ratio 11:7
Answer: Total parts = \(11 + 7 = 18\)
Suvidhea gets = \(\frac{11}{18} \times 5850 = 11 \times 325 = 3575\) Rs
Shweta gets = \(\frac{7}{18} \times 5850 = 7 \times 325 = 2275\) Rs
Check: \(3575 + 2275 = 5850\) ✓
In simple words: Add the ratio parts, then multiply each ratio part by the total amount divided by total parts.

Exam Tip: To divide in a given ratio, find total parts, then give each person: (their share/total parts) × total amount

Question 6a (Ratio and Proportion). Which is greater: 2:3 or 3:4?
Answer: \(2:3 = \frac{2}{3} ≈ 0.667\)
\(3:4 = \frac{3}{4} = 0.75\)
Since 0.75 > 0.667, the ratio \(3:4\) is greater
In simple words: Convert ratios to decimals or fractions and compare.

Exam Tip: To compare ratios, convert to fractions with a common denominator or to decimals.

Question 6b (Ratio and Proportion). Which is greater: 5:6 or 7:8?
Answer: \(5:6 = \frac{5}{6} ≈ 0.833\)
\(7:8 = \frac{7}{8} = 0.875\)
Since 0.875 > 0.833, the ratio \(7:8\) is greater
In simple words: Convert to decimals to compare easily.

Exam Tip: When numerator and denominator are close, the ratio is closer to 1 and thus greater.

Question 6c (Ratio and Proportion). Which is greater: 11:12 or 3:5?
Answer: \(11:12 = \frac{11}{12} ≈ 0.917\)
\(3:5 = \frac{3}{5} = 0.6\)
Since 0.917 > 0.6, the ratio \(11:12\) is greater
In simple words: Convert both ratios to decimals and compare the decimal values.

Exam Tip: Ratios with numerator and denominator very close in value are generally larger ratios.

Question 7 (Ratio and Proportion). Find 3 equivalent ratios of 8:14
Answer: First, simplify: \(8:14 = 4:7\) (by dividing by 2)
Three equivalent ratios:
1. \(4:7\) (simplified form)
2. \(8:14\) (original)
3. \(12:21\) (multiply by 3)
4. \(16:28\) (multiply by 4)
In simple words: Multiply or divide both parts of the ratio by the same number to get equivalent ratios.

Exam Tip: You can generate equivalent ratios by multiplying both parts by any positive number (2, 3, 4, 5, etc.)

Question 1 (Practical Geometry). Draw line segment AB = 5.4 cm. From this cut off AK = 3.8 cm
Answer: Steps:
1. Use a ruler and draw a straight line
2. Mark point A on the line
3. From A, measure 5.4 cm using the ruler and mark point B
4. From A, measure 3.8 cm using the ruler and mark point K
5. K lies between A and B on the line segment AB
6. KB = AB - AK = 5.4 - 3.8 = 1.6 cm
In simple words: Draw a line segment AB. Mark point K on it at distance 3.8 cm from A.

Exam Tip: Always use a ruler for accurate measurements. Mark points clearly and label them.

Question 2 (Practical Geometry). Draw MN = 3.1 cm and RS = 2.9 cm. Then draw AB = MN + RS
Answer: Steps:
1. Draw line segment MN = 3.1 cm using a ruler
2. Draw line segment RS = 2.9 cm using a ruler
3. AB = MN + RS = 3.1 + 2.9 = 6.0 cm
4. Draw a new line segment AB = 6.0 cm
5. You can verify by placing MN and RS end-to-end; they equal AB
In simple words: Draw two line segments separately, then add their lengths to draw a third segment equal to their sum.

Exam Tip: To add lengths graphically, you can place segments end-to-end or use the compass method.

Question 3 (Practical Geometry). Take line XY = 5.2 cm. Take point M on it. At M draw perpendicular MN to XY
Answer: Steps:
1. Draw line segment XY = 5.2 cm
2. Mark any point M on XY
3. Using a compass and straightedge (or set square):
- Place compass point at M and draw two arcs on either side
- From each arc point, draw intersecting arcs below
- Draw a line through M and the intersection point
4. This line MN is perpendicular to XY
5. Verify: Angle XMN = Angle YMN = 90°
In simple words: Draw a line segment and a point on it, then construct a perpendicular line at that point using geometric tools.

Exam Tip: A perpendicular makes a 90° angle. Use a compass and ruler, or a set square for accurate perpendiculars.

Question 4 (Practical Geometry). Construct an angle of 45 degrees using ruler and compasses
Answer: Steps:
1. Draw a ray AB
2. Construct a 90° angle using the compass and ruler (perpendicular to AB)
3. Construct a 45° angle by bisecting the 90° angle:
- Place compass at the vertex, mark equal arcs on both rays of the 90° angle
- From both arc points, draw intersecting arcs
- Draw a line from vertex through intersection; this bisects the angle
4. The angle between this bisector and ray AB is 45°
In simple words: First make a 90° angle, then bisect it to get 45°.

Exam Tip: 45° = 90° ÷ 2. Use angle bisection to divide any angle into two equal parts.

Question 5 (Practical Geometry). Draw an angle of 60 degrees. Draw its bisector
Answer: Steps:
1. Draw a ray AB
2. Using compass, construct a 60° angle:
- Keep compass width equal to the radius
- From point A, draw an arc intersecting AB at point C
- Without changing width, place compass at C and draw another arc
- Connect A to the intersection point; this creates a 60° angle
3. To bisect the 60° angle:
- Place compass at vertex and mark equal arcs on both rays
- From both arc points, draw intersecting arcs
- Draw a line through vertex and intersection
4. This bisector divides the 60° angle into two 30° angles
In simple words: Construct a 60° angle, then bisect it by finding equal intersection points from arcs on both sides.

Exam Tip: In an equilateral triangle, all angles are 60°. For bisecting, the arcs must be equidistant from the vertex on both rays.