Expressions using Letter-NumbersClass 7 Mathematics NCERT Solutions

To Find: Formulas for the perimeter of given regular polygons.

Let: The length of one side of the regular polygon be denoted by the letter-number '$s$'.

(a) Triangle with all sides equal (Equilateral Triangle): An equilateral triangle has 3 equal sides. Perimeter = Sum of all sides $P = s + s + s = 3s$ Formula: $P = 3s$

(b) A regular pentagon: A regular pentagon has 5 equal sides. Perimeter = Sum of all sides $P = s + s + s + s + s = 5s$ Formula: $P = 5s$

(c) A regular hexagon: A regular hexagon has 6 equal sides. Perimeter = Sum of all sides $P = s + s + s + s + s + s = 6s$ Formula: $P = 6s$

Given: Length of the first pipe = 20 m Length of the second pipe = $k$ m

To Find: An expression for the combined length of the pipe.

Solution: The combined length is the sum of the lengths of the two pipes. Combined length = (Length of first pipe) + (Length of second pipe) Expression = $20 + k$

Final Answer: The expression for the combined length of the pipe is $20 + k$ meters.

To Find: Complete the given table by finding the expression and total amount for each row.

Row 2: The expression and total amount are given: $6 \times 100 + 4 \times 20 + 3 \times 5 = 695$. From this, we can fill the number of notes:

Given: Fixed start-up time for the mill = 10 seconds Time to grind 1 kg of grain = 8 seconds Amount of grain to grind = $y$ kg

To Find: The expression for the total time taken.

Solution: The total time is the sum of the fixed start-up time and the variable grinding time.

This corresponds to the expression $10 + 8 \times y$.

Final Answer: The correct expression is (d) $10+8 \times y$.

Let: The number be represented by the letter-number '$x$'.

(a) 5 more than a number This means we add 5 to the number. Expression: $x + 5$

(b) 4 less than a number This means we subtract 4 from the number. Expression: $x - 4$

(c) 2 less than 13 times a number First, find '13 times a number': $13 \times x = 13x$. Then, '2 less than' this result means we subtract 2. Expression: $13x - 2$

(d) 13 less than 2 times a number First, find '2 times a number': $2 \times x = 2x$. Then, '13 less than' this result means we subtract 13. Expression: $2x - 13$

(a) Situation for $8 \times x+3 \times y$ (or $8x + 3y$) This expression represents a total value calculated from two different items with different quantities and prices. Example Situation: A person buys $x$ notebooks costing ₹8 each and $y$ pens costing ₹3 each. The total amount of money spent is given by the expression $8x + 3y$.

(b) Situation for $15 \times j-2 \times k$ (or $15j - 2k$) This expression represents a starting amount from which some other amount is subtracted. Example Situation: A fruit seller has 15 boxes, and each box contains $j$ apples. The total number of apples is $15j$. If the fruit seller sells 2 boxes, each containing $k$ oranges, this is not a valid situation as apples and oranges cannot be subtracted. A better situation is: A fruit seller has 15 boxes, each containing $j$ mangoes. From this stock, he sells $2k$ mangoes. The number of mangoes remaining is given by the expression $15j - 2k$.

Given: A $2 \times 3$ grid of dates from a calendar. The date in the bottom middle cell is '$w$'.

To Find: Expressions for the dates in the blank cells.

Final Answer: The expressions for the dates in the blank cells are:

Mistake Identification: The expression 3a was interpreted as concatenating the digit '3' with the value of 'a' (which is 3) to form the number 33.

Explanation of Error: The notation 3a in algebra means '3 multiplied by a' or 3 \times a.

Correction: Given $a=3$. $3a = 3 \times a$ $= 3 \times 3$ $= 9$

Correct Value: 9

Mistake Identification: The expression 5b was interpreted as concatenating the digit '5' with the value of 'b' (which is 4) to form the number 54. Then 2 was added: $54+2=56$.

Explanation of Error: The notation 5b means '5 multiplied by b' or 5 \times b.

Correction: Given $b=4$. $5b+2 = (5 \times b) + 2$ $= (5 \times 4) + 2$ $= 20 + 2$ $= 22$

Correct Value: 22

Mistake Identification: A simple subtraction error was made. $5-5$ was incorrectly calculated as 5.

Explanation of Error: Subtracting a number from itself results in 0.

Correction: Given $c=5$. $c-5 = 5 - 5$ $= 0$

Correct Value: 0

Mistake Identification: The expression d+e was evaluated by concatenating the values of d (2) and e (3) to form the number 23, instead of performing addition.

Explanation of Error: The '+' symbol represents the operation of addition.

Correction: Given $d=2$ and $e=3$. $d+e = 2 + 3$ $= 5$

Correct Value: 5

Mistake Identification: The expression x-y was evaluated by concatenating the values of x (6) and y (2) to form the number 62, instead of performing subtraction.

Explanation of Error: The '-' symbol represents the operation of subtraction.

Correction: Given $x=6$ and $y=2$. $x-y = 6 - 2$ $= 4$

Correct Value: 4

Mistake Identification: The expression km was interpreted as concatenating the values of k (3) and m (4) to form the number 34.

Explanation of Error: When two letter-numbers are written together like km, it implies multiplication: k \times m.

Correction: Given $k=3$ and $m=4$. $km = k \times m$ $= 3 \times 4$ $= 12$

Correct Value: 12

To Find: The value of the expression $23-10 \times 2$.

Solution: According to the order of operations (BODMAS/PEMDAS), multiplication is performed before subtraction. First, calculate the multiplication part: $10 \times 2 = 20$ Now, perform the subtraction: $23 - 20 = 3$

Final Answer: The value of the expression is 3.

To Find: The value of the expression $83+28-13+32$.

Solution: We can rearrange the terms to make the calculation easier (using commutative and associative properties of addition). Group the numbers that are easy to add or subtract. $83+28-13+32 = (83 - 13) + (28 + 32)$ First, calculate the values inside the brackets: $(83 - 13) = 70$ $(28 + 32) = 60$ Now, add the results: $70 + 60 = 130$

Final Answer: The value of the expression is 130.

To Find: The value of the expression $34-14+20$.

Solution: Performing the operations from left to right: First, perform the subtraction: $34 - 14 = 20$ Now, perform the addition: $20 + 20 = 40$

Final Answer: The value of the expression is 40.

To Find: The value of the expression $42+15-(8-7)$.

Solution: According to the order of operations, we solve the expression inside the brackets first. $(8-7) = 1$ Now the expression becomes: $42 + 15 - 1$ Performing the operations from left to right: $42 + 15 = 57$ $57 - 1 = 56$

Final Answer: The value of the expression is 56.

To Find: The value of the expression $68-(18+13)$.

Solution: According to the order of operations, we solve the expression inside the brackets first. $(18+13) = 31$ Now the expression becomes: $68 - 31$ Performing the subtraction: $68 - 31 = 37$

Final Answer: The value of the expression is 37.

To Find: The value of the expression $7 \times 4+9 \times 6$.

Solution: According to the order of operations, multiplication is performed before addition. First, calculate the multiplication parts: $7 \times 4 = 28$ $9 \times 6 = 54$ Now, perform the addition: $28 + 54 = 82$

Final Answer: The value of the expression is 82.

To Find: The value of the expression $20+8 \times (16-6)$.

Solution: According to the order of operations, we solve the expression inside the brackets first. $(16 - 6) = 10$ Now the expression becomes: $20 + 8 \times 10$ Next, perform the multiplication before addition: $8 \times 10 = 80$ Finally, perform the addition: $20 + 80 = 100$

Final Answer: The value of the expression is 100.