Q1. Find the value of 313824 by using prime factorization.
Q.2 Estimate the value of 63
Q.3 Determine whether each of the following is a prime or composite number
a. 753 b.757
Q.4 The number 840 and 8316 , written as the product of their prime factors, are
840 = 23 x 3 x 5 x7 and 8316 = 22 x 33 x 7 x 11. Hence, find
i) The greatest whole number that will divide both 840 and 8316 exactly,
ii) The smallest whole number that is divisible by both 840 and 8316
Q.5 The LCM of 6 , 12 and n is 660. Find all the possible values of n
Q.6 Ali needs to pack 108 stalks of roses, 81 stalk of lilies and 54 stalks orchids into identical baskets so that each type of flowers is equally distributed among the baskets. Find
i) the largest number of baskets that can be packed,
ii) the number of each type of flowers in a basket.
Answers
Answer:
ñGiven
Length of rectangle ⇒ 3x - 4y + 6z
Perimeter of rectangle ⇒ 7x + 8y + 17z
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To Find
The breadth of the rectangle.
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Solution
Perimeter of the given rectangle ⇒ 7x + 8y + 17z
Perimeter of rectangle ⇒ 2 × (Length + Breadth)
Length ⇒ 3x - 4y + 6z
Breadth ⇒ b
Equation to find the breadth ⇒ 2 (3x - 4y + 6z + b) = 7x + 8y + 17z
Let's solve your equation step-by-step
2 (3x - 4y + 6z + b) = 7x + 8y + 17z
Step 1: Open the bracket and substitute the values.
⇒ 2 (3x - 4y + 6z + b) = 7x + 8y + 17z
⇒ 6x - 8y + 12z + 2b = 7x + 8y + 17z
Step 2: Subtract 7x from both sides of the equation.
⇒ 6x - 8y + 12z + 2b - 7x = 7x + 8y + 17z
⇒ -x - 8y + 12z + 2b = 8y + 17z
Step 3: Subtract 8y from both sides of the equation.
⇒ -x - 8y + 12z + 2b - 8y = 8y + 17z - 8y
⇒ -x - 16y + 12z + 2b = 17z
Step 4: Subtract 17z from both sides of the equation.
⇒ -x - 16y + 12z + 2b - 17z = 17z - 17z
⇒ -x - 16y - 5z + 2b = 0
Step 5: Add x + 16y + 5z to both sides of the equation.
⇒ -x - 16y - 5z + 2b + x + 16y + 5z = x + 16y + 5z
⇒ 2b = x + 16y + 5z
Step 6: Divide 2 from both sides of the equation.
⇒ \dfrac{2b}{2} = \dfrac{x+16y+5z}{2}
2
2b
=
2
x+16y+5z
⇒ b= \dfrac{x+16y+5z}{2}b=
2
x+16y+5z
Let's verify the value of the breadth.
⇒ 2(3x-4y+6z +\dfrac{x+16y+5z}{2} ) = 7x + 8y + 17z2(3x−4y+6z+
2
x+16y+5z
)=7x+8y+17z
⇒ 2(3x-4y+6z +(x+16y+5z) \times \dfrac{1}{2} ) = 7x + 8y + 17z2(3x−4y+6z+(x+16y+5z)×
2
1
)=7x+8y+17z
⇒ 2(3x-4y+6z +\dfrac{1}{2} x+8y+\dfrac{5}{2} z ) = 7x + 8y + 17z2(3x−4y+6z+
2
1
x+8y+
2
5
z)=7x+8y+17z
⇒ 2(3x+\dfrac{1}{2} x+8y-4y +\dfrac{5}{2} z +6z) = 7x + 8y + 17z2(3x+
2
1
x+8y−4y+
2
5
z+6z)=7x+8y+17z
⇒ 2(\dfrac{6}{2} x+\dfrac{1}{2} x+8y-4y +\dfrac{5}{2} z +\dfrac{12}{2} z) = 7x + 8y + 17z2(
2
6
x+
2
1
x+8y−4y+
2
5
z+
2
12
z)=7x+8y+17z
⇒ 2(\dfrac{7}{2} x+4y +\dfrac{17}{2} z) = 7x + 8y + 17z2(
2
7
x+4y+
2
17
z)=7x+8y+17z
⇒ 7x + 8y + 17z = 7x + 8y + 17z7x+8y+17z=7x+8y+17z
∴ The value of the breadth of the rectangle is ⇒ \bf \dfrac{x+16y+5z}{2}
2
x+16y+5z
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