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Question

Hi, I would really appreciate it if I could get help with this question. 🙂

A rectangular board measures 630cm by 396cm. It is divided into small squares of equal size.

(a) Find the least number of squares.

(b) (i) Find the second largest possible length of the side of a square.

      (ii) Find the number of squares in this case.

Thank you!

Answer

(a)
(Note : To arrive at the least number of squares, we have to find the largest possible length of each square that would fit.)
630 = 2 x 3^2 x 5 x 7
396 = 2^2 x 3^2 x 11
HCF = 2 x 3^2 = 18
Largest possible length of each square = 18 cm
630/18 = 35
396/18 = 22
35 x 22 = 770
Least number of squares = 770

(b)(i)
Second largest possible length of the side of a square = 3^2 = 9 cm

(ii)
Number of squares in this case = (630 x 396)/81 = 3080

Ans : (a) 770 squares; (b)(i) 9 cm, (ii) 3080 squares.

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Thank you!

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Hello Rae! 

When you encounter questions that ask you for “the most” or “the least” possible options, they usually require you to find the Highest Common Factor (HCF) or Lowest Common Multiple (LCM) of 2 or more integers.

(a) to find the least, we must divide 630cm & 396cm by the largest possible factor.

To find HCF of 630, you must begin by dividing it with its largest prime number (in this case, 7) then dividing by lower prime numbers until it is fully divided.

630 = 2 x 3² x 5 x 7
396 = 2²x 3² x 11

Then, you multiply the lowest common factors
2 x 3² = 18 (H.C.F)

This is the largest possible length of a square. The second largest is found by dividing by 2.
3² = 9cm. (b)(i)

630cm ÷ 18cm = 35 squares
396cm ÷ 18cm = 22 squares

The retangular board can thus be cut into 35 squares length-wise, and 22 squares width-wise.

35 x 22 = 770 squares at least (a)

With a 9cm length, each square would be 9²= 81cm²

Take the board’s area divided by the squares’ area.
630cm x 396cm = 249480cm²
249480cm² ÷ 81cm² = 3080 squares (b)(ii)

Hope this helps you understand the question better! If you need help, just let us know 🙂

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Thank you!

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