Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 2) | Concept of Maths

Question 1

1. A gardener is planting marigolds in a triangular arrangement. The first row has 3 plants, the second has 7 plants, and the third has 11 plants. If this arithmetic pattern continues, what is the rule for the number of plants in the $n$-th row?

A) $3n + 4$

B) $4n - 1$

C) $4n + 3$

D) $3n - 1$

Answer

Correct Answer: B

Solution: Row 1 ($n=1$): $3$ plants. Row 2 ($n=2$): $7$ plants. Row 3 ($n=3$): $11$ plants. The difference between consecutive rows is $7 - 3 = 4$. The rule follows the form $4n + c$. For $n=1$, $4(1) + c = 3 \Rightarrow c = -1$. Thus, the rule is $4n - 1$.

Question 2

2. To make a chain of $x$ triangles using matchsticks, where each triangle shares one side with the next, the number of matchsticks required follows a pattern. 1 triangle needs 3 matches, 2 triangles need 5 matches, and 3 triangles need 7 matches. What is the expression for $x$ triangles?

A) $2x + 1$

B) $3x$

C) $x + 2$

D) $2x - 1$

Answer

Correct Answer: A

Solution: Let's list the matches: $x=1 \to 3$, $x=2 \to 5$, $x=3 \to 7$. This is a sequence increasing by 2. The formula is $2x + 1$. For $x=1$, $2(1)+1=3$. For $x=2$, $2(2)+1=5$.

Question 3

3. A local taxi service in Ranchi charges a fixed amount of $₹ 500$ as a booking fee and an additional $₹ 15$ for every kilometer ($d$) traveled. Which expression represents the total cost $C$?

A) $C = 500d + 15$

B) $C = 515d$

C) $C = 15d + 500$

D) $C = 15(d + 500)$

Answer

Correct Answer: C

Question 4

4. Isha is designing a walkway with square tiles. Section 1 uses 5 tiles, Section 2 uses 9 tiles, and Section 3 uses 13 tiles. Which expression describes the number of tiles in Section $n$?

A) $5n$

B) $n + 4$

C) $5n - 1$

D) $4n + 1$

Answer

Correct Answer: D

Solution: The sequence is $5, 9, 13...$. The common difference is $4$. The rule is $4n + c$. Since $4(1) + c = 5$, then $c = 1$. The rule is $4n + 1$.

Question 5

5. A software engineer has a starting salary of $S$. Every year, she receives a fixed increment of $I$. What will be her salary after $y$ years?

A) $S + I$

B) $S + (y \times I)$

C) $y(S + I)$

D) $S - yI$

Answer

Correct Answer: B

Solution: After 1 year, salary = $S + I$. After 2 years, salary = $S + 2I$. After $y$ years, the total increment is $y \times I$. So, salary = $S + yI$.

Question 6

6. Consider a row of $k$ identical square tables pushed together. One table seats 4 people. Two tables seat 6 people. Three tables seat 8 people. How many people can be seated around $k$ tables?

A) $2k + 2$

B) $4k$

C) $3k + 1$

D) $2k + 4$

Answer

Correct Answer: A

Solution: Sequence: $4, 6, 8...$. The difference is 2. The formula is $2k + c$. For $k=1$, $2(1) + c = 4 \Rightarrow c = 2$. The rule is $2k + 2$. Alternatively, 2 people sit at the ends, and $2$ people sit at each table along the sides ($2 \times k$). Total = $2k + 2$.

Question 7

7. Observations show that the sum of the first $n$ odd natural numbers follows a specific pattern: $1 = 1^2$, $1+3 = 4 = 2^2$, $1+3+5 = 9 = 3^2$. What is the sum of the first $n$ odd numbers?

A) $2n - 1$

B) $n + 1$

C) $n^2$

D) $n(n+1)$

Answer

Correct Answer: C

Solution: The pattern clearly shows that the sum of the first $n$ odd numbers is the square of $n$. Sum = $n \times n = n^2$.

Question 8

8. A baker prepares $b$ batches of cookies. Each batch contains 12 cookies. If the baker eats 5 cookies from the total amount, which expression represents the cookies remaining?

A) $12 + b - 5$

B) $5b - 12$

C) $12(b - 5)$

D) $12b - 5$

Answer

Correct Answer: D

Solution: Total cookies produced = $12 \times b$ (12 cookies per batch). Cookies remaining after eating 5 = Total - 5 = $12b - 5$.

Question 9

9. A wooden ladder has rungs spaced $25$ cm apart. If there are $r$ rungs, the distance from the first rung to the $r$-th rung is given by which pattern?

A) $25r$

B) $25(r - 1)$

C) $25r + 25$

D) $r + 25$

Answer

Correct Answer: B

Solution: The number of gaps between $r$ rungs is $(r-1)$. Since each gap is $25$ cm, the total distance is $25 \times (r-1)$.

Question 10

10. Aarav saves money every week. In week 1 he saves $₹ 50$. In week 2 he saves $₹ 80$. In week 3 he saves $₹ 110$. If the pattern continues, how much does he save in week $w$?

A) $30w + 20$

B) $30w + 50$

C) $50w + 30$

D) $80w - 30$

Answer

Correct Answer: A

Solution: The savings are $50, 80, 110...$. The common difference is $30$. Rule: $30w + c$. For $w=1$, $30(1) + c = 50 \Rightarrow c = 20$. The expression is $30w + 20$.

Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 2) | Concept of Maths