CBSE Class 6 Mathematics Algebra Worksheet Set 05

Question 1. An aeroplane goes 10 Km in 1 minute. Find the distance covered by the aeroplane in x minutes.
Answer: Distance covered in 1 minute = 10 km
Distance covered in \(x\) minutes = \(10 \times x = 10x\) km
In simple words: Multiply the speed per minute by the number of minutes to get the total distance.

Exam Tip: When forming algebraic expressions for rates, multiply the given rate by the variable representing time or quantity.

Question 2. How many sticks will be used to make the alphabet E ?
Answer: To form the letter E using straight line segments (sticks), we need 1 vertical stick and 3 horizontal sticks.
Total number of sticks = \(1 + 3 = 4\)
In simple words: Count the individual straight line segments needed to draw the letter E.

Exam Tip: Break down geometric letters or shapes into individual line segments to count the exact number of matchsticks required.

Question 3. Write the algebraic expression for the following -
a) Shyam has x pens. Ram has 15 more pens than Shyam. How many pens to pens Shyam have _________________.
Answer: Number of pens Shyam has = \(x\)
Ram has 15 more pens than Shyam, so Ram's pens = \(x + 15\)
In simple words: The word 'more' indicates addition, so add 15 to Shyam's pens.

Exam Tip: Keywords like 'more than' or 'increased by' always translate to addition in algebra.

Question 3. b) 21 added to 3x = ______________________.
Answer: \(3x + 21\)
In simple words: Add 21 to the term \(3x\).

Question 3. c) 45 subtracted from 5x _______________
Answer: \(5x - 45\)
In simple words: Subtract 45 from \(5x\), which means 45 comes after the minus sign.

Exam Tip: Pay close attention to the order in subtraction: 'subtracted from' means the second quantity is taken away from the first.

Question 3. d) 9 times z from which 5 is subtracted.
Answer: \(9z - 5\)
In simple words: First write 9 times \(z\) as \(9z\), then subtract 5 from it.

Exam Tip: 'Times' denotes multiplication, written tightly as a coefficient with a variable.

Question 3. e) 10 added to the quotient of x and 5.
Answer: \(\frac{x}{5} + 10\)
In simple words: Quotient means division, so divide \(x\) by 5 and add 10.

Exam Tip: 'Quotient of \(a\) and \(b\)' means division in the exact order: \(\frac{a}{b}\).

Question 3. f) Quotient of X and Y added to the product of a and b. ________________
Answer: \(ab + \frac{X}{Y}\)
In simple words: Product means multiplication (\(ab\)) and quotient means division (\(X/Y\)), then add them together.

Exam Tip: Product implies multiplication, quotient implies division.

Question 3. g) 5 times a number x is equal to 4 less than a number y ________________
Answer: \(5x = y - 4$
In simple words: 5 times \(x\) is \(5x\), and 4 less than \(y\) is \(y - 4\), set them equal.

Exam Tip: 'Is equal to' translates directly to the equals sign (=).

Question 3. i) 8 subtracted from x gives 1 ____________________
Answer: \(x - 8 = 1$
In simple words: Take 8 away from \(x\) and set the result equal to 1.

Exam Tip: The word 'gives' or 'results in' represents the equals sign.

Question 3. j) y divided by 4 added to 8 gives 1 ____________________
Answer: \(\frac{y}{4} + 8 = 1\)
In simple words: Divide \(y\) by 4, add 8, and equate to 1.

Exam Tip: Translate each phrase sequentially from left to right as you read the sentence.

Question 4. Solve
a) x + 3 = 10
Answer: \(x + 3 = 10$
\(x = 10 - 3$
\(x = 7$
In simple words: Transpose +3 to the right side where it becomes -3.

Exam Tip: When moving a term across the equals sign, always change its sign (positive to negative, negative to positive).

Question 4. b) 5x = 25
Answer: \(5x = 25$
\(x = \frac{25}{5}
\(x = 5$
In simple words: Divide both sides by 5 to isolate \(x\).

Exam Tip: Multiplication becomes division when transposed across the equals sign.

Question 4. c) x - 4 = 7
Answer: \(x - 4 = 7$
\(x = 7 + 4$
\(x = 11$
In simple words: Move -4 to the right side to make it +4.

Exam Tip: Always perform inverse operations to solve for the unknown variable.

Question 4. d) \(\frac{x}{8} = 2$
Answer: \(\frac{x}{8} = 2\)
\(x = 2 \times 8$
\(x = 16$
In simple words: Multiply both sides by 8 to cancel out the denominator.

Exam Tip: Division in an equation is solved by multiplying both sides by the divisor.

Question 4. e) 5x - 8 = 2
Answer: \(5x - 8 = 2$
\(5x = 2 + 8$
\(5x = 10$
\(x = \frac{10}{5} = 2$
In simple words: First add 8 to both sides, then divide by 5.

Exam Tip: Solve two-step equations by isolating the term with the variable first, then isolating the variable itself.

Question 5. Complete the table using the given formula.
a) x = y + 5

Answer: Using \(y = x - 5\):
For \(x = 2\), \(y = 2 - 5 = -3$
For \(x = 3\), \(y = 3 - 5 = -2$
For \(x = 4\), \(y = 4 - 5 = -1$
For \(x = 5\), \(y = 5 - 5 = 0$
For \(x = 6\), \(y = 6 - 5 = 1$
In simple words: Rearrange the formula to find \(y\) by subtracting 5 from each given value of \(x\).

Exam Tip: Substitute each column value one by one into the rearranged equation to avoid calculation errors.

Question 5. b) a - 7

Answer: Let \(b = a - 7\):
For \(a = 10\), \(b = 10 - 7 = 3$
For \(a = 9\), \(b = 9 - 7 = 2$
For \(a = 8\), \(b = 8 - 7 = 1$
For \(a = 7\), \(b = 7 - 7 = 0$
For \(a = 11\), \(b = 11 - 7 = 4$
In simple words: Subtract 7 from each given value of \(a\) to find the corresponding value of \(b\).

Exam Tip: Check your calculations by working backwards from the completed row.

Question 6. Write the next number in each number patterns.
a) 7, 13, 19, 25, ___, ___, ___
Answer: 31, 37, 43
In simple words: The pattern increases by 6 each time (\(13 - 7 = 6\)). Add 6 successively.

Exam Tip: Find the common difference between consecutive terms to identify arithmetic patterns.

Question 6. b) 4, 9, 16, ___, ___, ___
Answer: 25, 36, 49
In simple words: These are squares of natural numbers (\(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\)). Next are \(5^2 = 25\), \(6^2 = 36\), \(7^2 = 49\).

Exam Tip: Recognize square number sequences instantly up to at least 20.

Question 6. c) 70, 56, 42, ___, ___, ___
Answer: 28, 14, 0
In simple words: The numbers decrease by 14 each time (\(70 - 56 = 14\)). Subtract 14 successively.

Exam Tip: For decreasing patterns, calculate the difference between the first two terms and subtract it continuously.

Question 6. d) 501, 491, 481, ___, ___, ___
Answer: 471, 461, 451
In simple words: The pattern decreases by 10 each time. Subtract 10 successively.

Exam Tip: Look at the tens and units digits to quickly spot subtraction of powers of 10.

Question 7. A number y is always 10 less than a number x
i) Write down the rule which connects x and y.
Answer: \(y = x - 10$
In simple words: Since \(y\) is 10 less than \(x\), subtract 10 from \(x\) to get \(y\).

Exam Tip: 'Less than' means subtraction from the reference variable.

Question 7. ii) Find y when x is 19.
Answer: Substituting \(x = 19\) into \(y = x - 10\):
\(y = 19 - 10 = 9$
In simple words: Replace \(x\) with 19 in the rule formula.

Exam Tip: Always state the formula before substituting given numerical values.

Question 8. Find the ratio for the following in the lowest terms
a) 96 m to 5 km
Answer: Convert km to meters: \(5 \text{ km} = 5 \times 1000 = 5000 \text{ m}$
Ratio = \(\frac{96}{5000}\)
Dividing numerator and denominator by their HCF (32):
\(\frac{96 \div 32}{5000 \div 32} = \frac{3}{156.25}\)
Alternatively, dividing by 8: \(\frac{12}{625}\)
Ratio = 12 : 625
In simple words: Convert both quantities to the same unit (meters) first, then simplify the fraction.

Exam Tip: Quantities must be in the same units before finding their ratio.

Question 8. b) Rs. 9 to 1800 paise
Answer: Convert rupees to paise: Rs. \(9 = 9 \times 100 = 900 \text{ paise}$
Ratio = \(\frac{900}{1800} = \frac{1}{2}\)
Ratio = 1 : 2
In simple words: Convert Rs. 9 into 900 paise, then simplify the fraction \(900/1800\).

Exam Tip: Always convert larger currency units into smaller units (rupees to paise) to avoid decimals.

Question 8. c) 45 minutes to 2 hours
Answer: Convert hours to minutes: \(2 \text{ hours} = 2 \times 60 = 120 \text{ minutes}$
Ratio = \(\frac{45}{120}\)
Dividing numerator and denominator by 15: \(\frac{45 \div 15}{120 \div 15} = \frac{3}{8}\)
Ratio = 3 : 8
In simple words: Convert 2 hours into 120 minutes, then reduce \(45/120\) to lowest terms by dividing by 15.

Exam Tip: Find the HCF of both numbers to reduce ratios in a single step.

Question 8. d) 70 g to 2.5 kg
Answer: Convert kg to grams: \(2.5 \text{ kg} = 2.5 \times 1000 = 2500 \text{ g}$
Ratio = \(\frac{70}{2500} = \frac{7}{250}\)
Ratio = 7 : 250
In simple words: Convert 2.5 kg to 2500 grams, then simplify \(70/2500\) by cancelling a zero.

Question 9. Find the value of x
a) 14 : 16 = x : 8
Answer: Product of extremes = Product of means
\(14 \times 8 = 16 \times x$
\(112 = 16x$
\(x = \frac{112}{16} = 7$
In simple words: Use cross multiplication: product of outer terms equals product of inner terms.

Exam Tip: For proportions (\(a:b :: c:d\)), remember that \(ad = bc\).

Question 9. b) 4 : x = 20 : 45
Answer: \(4 \times 45 = x \times 20$
\(180 = 20x$
\(x = \frac{180}{20} = 9$
In simple words: Cross multiply to get \(20x = 180\), then solve for \(x\).

Exam Tip: Keep the variable term on one side and divide by its coefficient.

Question 9. c) 2 : 7 :: x : 28
Answer: \(2 \times 28 = 7 \times x$
\(56 = 7x$
\(x = \frac{56}{7} = 8$
In simple words: Multiply outer numbers (\(2 \times 28 = 56\)) and divide by 7.

Exam Tip: Proportions state that two ratios are equal.

Question 9. d) x : 3 :: 52 : 39
Answer: \(x \times 39 = 3 \times 52$
\(39x = 156$
\(x = \frac{156}{39} = 4$
In simple words: Cross multiply \(x \times 39 = 3 \times 52\) to find \(x\).

Exam Tip: Simplify large products before dividing to make mental math easier.

Question 10. A car travels 360 km in 12 litres of petrol.
i) How much distance will it travel in 20 litres of petrol ?
Answer: Distance traveled in 12 litres = 360 km
Distance traveled in 1 litre = \(\frac{360}{12} = 30 \text{ km}\)
Distance traveled in 20 litres = \(30 \times 20 = 600 \text{ km}$
In simple words: Find the distance per litre first (unitary method), then multiply by 20.

Exam Tip: Always find the value of a single unit first in direct variation problems.

Question 10. ii) How much petrol is required for travelling 20 km. ?
Answer: Petrol required for 360 km = 12 litres
Petrol required for 1 km = \(\frac{12}{360} = \frac{1}{30} \text{ litres}\)
Petrol required for 20 km = \(\frac{1}{30} \times 20 = \frac{20}{30} = \frac{2}{3} \text{ litres}\)
In simple words: Divide the total petrol by the total distance to find petrol per km, then multiply by 20.

Question 11. Divide 420 g in the ratio of 2:3.
Answer: Sum of ratio terms = \(2 + 3 = 5$
First part = \(\frac{2}{5} \times 420 = 2 \times 84 = 168 \text{ g}\)
Second part = \(\frac{3}{5} \times 420 = 3 \times 84 = 252 \text{ g}\)
In simple words: Add ratio parts to find the total (5), divide 420 by 5 to get 84, then multiply by 2 and 3 respectively.

Exam Tip: Always check your answer by adding the divided parts to ensure they sum up to the original total (\(168 + 252 = 420\)).

Question 12. A total 2400 candidates appeared in an examination. If the ratio of males to females is 2 : 6, find the number of males and females.
Answer: Simplified ratio of males to females = \(2 : 6 = 1 : 3$
Sum of ratio terms = \(1 + 3 = 4$
Number of males = \(\frac{1}{4} \times 2400 = 600\)
Number of females = \(\frac{3}{4} \times 2400 = 1800\)
In simple words: Simplify the ratio 2:6 to 1:3, divide 2400 by 4 parts (600 each), and assign 1 part to males and 3 parts to females.

Exam Tip: Simplifying ratios before solving makes calculations much simpler.

Question 13. Rachna purchased 14 pens for Rs. 35 and Raj purchased 12 pens for Rs. 30. Who bought more expensive pens ?
Answer: Cost of 1 pen for Rachna = \(\frac{35}{14} = \frac{5}{2} = \text{Rs. } 2.50\)
Cost of 1 pen for Raj = \(\frac{30}{12} = \frac{5}{2} = \text{Rs. } 2.50\)
Both bought pens at the same price of Rs. 2.50 per pen. Neither bought more expensive pens.
In simple words: Find the unit price (cost per single pen) for both Rachna and Raj and compare them.

Exam Tip: To compare unit costs, always calculate the price of exactly one item.

Question 14. The ratio of racquets to shuttle cock is 14 : 7. If the racquets are 152, find the number of shuttle cocks.
Answer: Simplified ratio = \(14 : 7 = 2 : 1$
Let number of shuttle cocks be \(x$.
\(\frac{\text{Racquets}}{\text{Shuttle cocks}} = \frac{2}{1}\)
\(\frac{152}{x} = \frac{2}{1}\)
\(2x = 152$
\(x = \frac{152}{2} = 76$
Number of shuttle cocks = 76
In simple words: Set up a proportion where 2 parts correspond to 152 racquets, meaning each part is 76.

Exam Tip: Always reduce ratios to lowest terms before setting up proportions.

Question 15. What fraction of a revolution does an hour hand of a clock turns when it goes from
i) 5 to 8 in clockwise direction
Answer: Hours covered from 5 to 8 = \(8 - 5 = 3 \text{ hours}$
Total hours on a clock face = 12
Fraction = \(\frac{3}{12} = \frac{1}{4}\)
In simple words: 3 hours out of 12 hours represents one-quarter of a full clock revolution.

Exam Tip: A clock face is divided into 12 equal hour intervals.

Question 15. ii) 2 to 8 in anticlock wise direction
Answer: Moving anticlockwise from 2 to 8 covers 6 hours.
Fraction = \(\frac{6}{12} = \frac{1}{2}\)
In simple words: 6 hours out of 12 hours is exactly half of a revolution.

Exam Tip: Anticlockwise means moving in the reverse direction of clock hands (counter-clockwise).

Question 16. Where will the hand of a clock stop if it
i) starts at 6 and makes \(\frac{1}{2}\) of a revolution clockwise ?
Answer: \(\frac{1}{2}\) of a revolution = \( \frac{1}{2} \times 12 = 6 \text{ hours}$
Starting at 6 and moving 6 hours forward: \(6 + 6 = 12$
The hand will stop at 12.
In simple words: Add 6 hours to the starting position 6 to land on 12.

Exam Tip: Half a revolution equals 6 hours on a clock.

Question 16. ii) starts at 6 and makes \(\frac{1}{4}\) of a revolution anticlockwise?
Answer: \(\frac{1}{4}\) of a revolution = \( \frac{1}{4} \times 12 = 3 \text{ hours}$
Starting at 6 and moving 3 hours backward (anticlockwise): \(6 - 3 = 3$
The hand will stop at 3.
In simple words: Subtract 3 hours from 6 since it is moving anticlockwise.

Question 17. Which direction will you face if you start facing
a) North and make \(1\frac{1}{2}\) of a revolution clockwise.
Answer: \(1\frac{1}{2}\) revolutions mean 1 full complete circle plus a half-circle (\(180^\circ\)).
Starting facing North, 1 full revolution brings you back to North, and a further half-revolution points you directly opposite.
You will face South.
In simple words: A full revolution brings you back to your starting point, and the extra half turn points in the opposite direction.

Exam Tip: Full revolutions (360 degrees) bring you back to the exact same starting direction.

Question 17. b) South and make \(\frac{3}{4}\) of a revolution clockwise.
Answer: \(\frac{3}{4}\) of a revolution = \(270^\circ$ clockwise rotation.
Starting facing South, a \(270^\circ\) clockwise turn passes through West, North, and stops at East.
You will face East.
In simple words: Turning 3 quarters clockwise from South points you towards East.

Exam Tip: Each quarter turn on a compass represents 90 degrees or 3 hours on a clock face.

Question 18. Find the number of right angles turned by an hour hand of a clock when it goes from
a) 3 to 9
Answer: Hours passed from 3 to 9 = \(9 - 3 = 6 \text{ hours}$
Each right angle corresponds to 3 hours on a clock face.
Number of right angles = \( \frac{6}{3} = 2 \text{ right angles}$
In simple words: 6 hours represents 2 right angles (\(90^\circ + 90^\circ = 180^\circ\)).

Exam Tip: 3 hours on a clock equals 1 right angle (90 degrees).

Question 18. b) 1 to 10
Answer: Hours passed from 1 to 10 (clockwise) = 9 hours.
Number of right angles = \( \frac{9}{3} = 3 \text{ right angles}$
In simple words: 9 hours equals 3 right angles (\(270^\circ\)).

Exam Tip: Divide the number of hour intervals by 3 to find right angles.

Question 19. Classify the angles and triangles given below
Answer:
1. Triangle with side lengths 5 cm, 5 cm, 5 cm: Equilateral Triangle (all sides equal).
2. Angle measuring \(60^\circ$: Acute Angle.
3. Triangle with angles \(48^\circ\), \(110^\circ\), \(32^\circ$: Obtuse-angled Triangle (since one angle is \(110^\circ\)).
4. Triangle with side lengths 2 cm, 3 cm, 4 cm: Scalene Triangle (all sides unequal).
5. Triangle with a right-angle marker (\(90^\circ\)): Right-angled Triangle.
In simple words: Triangles are classified by their sides (equilateral, scalene) or angles (acute, obtuse, right).

Exam Tip: Always look for angle degree numbers and tick marks on sides to correctly classify triangles.

Question 20. What is common between
a) a rhombus and a square
Answer: Both have all four sides equal in length, and both have opposite sides parallel.
In simple words: Both are quadrilaterals with four equal sides.

Exam Tip: A square is a special type of rhombus where every interior angle is a right angle.

Question 20. b) a parallelogram and a rectangle
Answer: Both have opposite sides equal and parallel, and both have opposite angles equal with diagonals bisecting each other.
In simple words: Both share the properties of parallel opposite sides and equal opposite angles.

Exam Tip: A rectangle is a special parallelogram with 90-degree corners.

Question 21. How many degrees are there in \(\frac{2}{3}\) of a right angle.
Answer: One right angle = \(90^\circ$
\(\frac{2}{3}\) of a right angle = \( \frac{2}{3} \times 90^\circ = 2 \times 30^\circ = 60^\circ$
In simple words: Multiply \(90^\circ\) by \(2/3\) to get \(60^\circ\).

Exam Tip: Always substitute 90 degrees for 'a right angle' in degree calculations.

Question 22. How many diagonals does
a) a pentagon have
Answer: A pentagon has 5 sides (\(n = 5\)).
Formula for diagonals = \(\frac{n(n - 3)}{2}\)
Number of diagonals = \( \frac{5(5 - 3)}{2} = \frac{5 \times 2}{2} = 5 $
In simple words: A pentagon has 5 diagonals connecting non-adjacent vertices.

Exam Tip: Memorize the diagonal formula \(n(n-3)/2\) for any \(n\)-sided polygon.

Question 22. b) How many faces does a tetrahedron have ?
Answer: A tetrahedron is a triangular pyramid which has 4 triangular faces.
In simple words: A tetrahedron has 4 flat triangular faces.

Exam Tip: A tetrahedron is the simplest 3D polyhedron.

Question 23. Draw rectangle sketch of a
i) triangular Pyramid ii) a triangular Prism
Answer:
1. Triangular Pyramid: A 3D shape with a triangular base and 3 triangular faces meeting at a top apex point.
2. Triangular Prism: A 3D shape with two parallel triangular bases connected by 3 rectangular side faces.
In simple words: A pyramid comes to a single top point, while a prism has identical triangular ends connected by rectangles.

Exam Tip: Practice drawing 3D solids by drawing parallel transparent edges using dashed lines for hidden parts.