Chapter 4

3 Solution : Correct answer is (c).

The solution of which of the following equations is neither a fraction nor an integer?

• (a)2x + 6 = 0
• (b)3x – 5 = 0
• (c)5x – 8 = x + 4
• (d)4x + 7 = x + 2

The equation which cannot be solved in integers is

• (a)5y – 3 = – 18
• (b)3x – 9 = 0
• (c)3z + 8 = 3 + z
• (d)9y + 8 = 4y – 7

If 7x + 4 = 25, then x is equal to 29 100

• (a)
• (b)
• (c)2
• (d)3

7 7. The solution of the equation 3x + 7 = – 20 is 17 13

• (a)
• (b)– 9
• (c)9
• (d)7 3

The value of y for which the expressions (y – 15) and (2y + 1) become equal is

• (a)0
• (b)16
• (c)8
• (d)– 16 Words Numbers Algebra You can subtract the same number from both 4 + 7 = 11 x = y implies sides of an equation, – 3 = –3 x–z=y–z and the statement will 4+4= 8 still be true.

If k + 7 = 16, then the value of 8k – 72 is

• (a)0
• (b)1
• (c)112
• (d)56

If 43m = 0.086, then the value of m is

• (a)0.002
• (b)0.02
• (c)0.2
• (d)2 15-04-2018

x exceeds 3 by 7, can be represented as

• (a)x + 3 = 2
• (b)x + 7 = 3
• (c)x – 3 = 7
• (d)x – 7 = 3

The equation having 5 as a solution is:

• (a)4x + 1 = 2
• (b)3 – x = 8
• (c)x – 5 = 3
• (d)3 + x = 8

The equation having – 3 as a solution is:

• (a)x + 3 =1
• (b)8 + 2x = 3
• (c)10 + 3x = 1
• (d)2x + 1 = 3

Which of the following equations can be formed starting with x = 0 ?

• (a)2x + 1 = – 1
• (b)+ 5 = 7
• (c)3x – 1 = – 1
• (d)3x – 1 = 1

Which of the following equations cannot be formed using the equation x = 7?

• (a)2x + 1 =15
• (b)7x – 1 = 50
• (c)x – 3 = 4
• (d)–1=0

If = 3, then the value of 3x + 2 is

• (a)20
• (b)11
• (c)
• (d)8

Which of the following numbers satisfy the equation –6 + x = –12 ?

• (a)2
• (b)6
• (c)– 6
• (d)– 2

Shifting one term from one side of an equation to another side with a change of sign is known as

• (a)commutativity
• (b)transposition
• (c)distributivity
• (d)associativity One-step equations can be solved by applying a single inverse operation. To solve two-step equations, apply more than one inverse operation. The order of operations for 2x + 5 = 7 is to start with x, multiply by 2 and add 5. The result is 7. Order of Operations: x multiply by 2 add 5 7 To solve the equation, inverse the steps. Start with 7, subtract 5, then divide by 2 to find x. Solve the equation: 7 subtract 5 divide by 2 x 15-04-2018 In Questions 19 to 48, fill in the blanks to make the statements true.

The sum of two numbers is 60 and their difference is 30.

• (a)If smaller number is x, the other number is .(use sum)
• (b)The difference of numbers in term of x is .
• (c)The equation formed is .
• (d)The solution of the equation is . (e) The numbers are and . Words Numbers Algebra Multiply both sides of 2×3=6 x = y implies an equation by the same non-zero number, and 4×2×3=4×6 zx = zy (z ≠ 0) the statement will still be true. 8 × 3 = 24

Sum of two numbers is 81. One is twice the other.

• (a)If smaller number is x, the other number is .
• (b)The equation formed is .
• (c)The solution of the equation is .
• (d)The numbers are and .

In a test Abha gets twice the marks as that of Palak. Two times Abha's marks and three times Palak's marks make 280.

• (a)If Palak gets x marks, Abha gets marks.
• (b)The equation formed is .
• (c)The solution of the equation is .
• (d)Marks obtained by Abha are .

The length of a rectangle is two times its breadth. Its perimeter is 60 cm.

• (a)If the breadth of rectangle is x cm, the length of the rectangle is . 15-04-2018
• (b)Perimeter in terms of x is .
• (c)The equation formed is .
• (d)The solution of the equation is .

In a bag there are 5 and 2 rupee coins. If they are equal in number and their worth is 70, then

• (a)The worth of x coins of 5 each .
• (b)The worth of x coins of 2 each .
• (c)The equation formed is .
• (d)There are 5 rupee coins and 2 rupee coins. 1. Describe how you would solve 4 (x – 2) = 16. 2. Express how to check your solution to an equation.

In a Mathematics quiz, 30 prizes consisting of 1st and 2nd prizes only are to be given. 1st and 2nd prizes are worth 2000 and 1000, respectively. If the total prize money is 52,000 then show that:

• (a)If 1st prizes are x in number the number of 2nd prizes are .
• (b)The total value of prizes in terms of x are .
• (c)The equation formed is .
• (d)The solution of the equation is . (e) The number of 1st prizes are and the number of 2nd prizes are .

If z + 3 = 5, then z = .

is the solution of the equation 3x – 2 =7.

is the solution of 3x + 10 = 7.

If 2x + 3 = 5, then value of 3x + 2 is .

In integers, 4x – 1 = 8 has solution. 15-04-2018 Words Numbers Algebra You can divide both sides 4 × 3 = 12 x = y, implies of an equation by the 4 × 3 12 x y same non-zero number, = = ,z ≠ 0 2 2 z z and the statement will still be true. 12 =6

In natural numbers, 4x + 5 = – 7 has solution.

In natural numbers, x – 5 = – 5 has solution.

In whole numbers, x + 8 = 12 – 4 has solution.

If 5 is added to three times a number, it becomes the same as 7 is subtracted from four times the same number. This fact can be represented as .

x + 7 = 10 has the solution .

x – 0 = ; when 3x = 12.

x – 1= ; when 2x = 2.

x – = 15; when =6.

The solution of the equation x + 15 = 19 is .

Finding the value of a variable in a linear equation that the equation is called a of the equation.

Any term of an equation may be transposed from one side of the equation to the other side of the equation by changing the of the term. 9 18

If x= , then x = . 5 5

If 3 – x = – 4, then x = . 1 1

If x − = − , then x = . 2 2 15-04-2018 1 1

If − x = , then x = . 6 6

If 10 less than a number is 65, then the number is .

If a number is increased by 20, it becomes 45. Then the number is .

If 84 exceeds another number by 12, then the other number is . 7 7

If x − = , then x = . 8 8 Give two words or phrases that can be used to express each operation: addition, subtraction, multiplication, and division. Express 5 + 7n in words in at least two different ways. In Questions 49 to 55, state whether the statements are True or False.

5 is the solution of the equation 3x + 2 = 17.

is the solution of the equation 4x – 1 = 8.

4x – 5 = 7 does not have an integer as its solution.

One third of a number added to itself gives 10, can be represented as + 10 = x .

is the solution of the equation 8x – 5 = 7.

If 4x – 7 = 11, then x = 4. 5x − 7

If 9 is the solution of variable x in the equation = y , then the value of y is 28. 15-04-2018

Match each of the entries in Column I with the appropriate entries in Column II. Column I Column II (i) x+5=9

• (A)− (ii) x–7=4
• (B)(iii) =–5
• (C)4 (iv) 5x = 30
• (D)6 (v) The value of y which satisfies 3y = 5 (E) 11 (vi) If p = 2, then the value of (1 – 3p) (F) – 60 (G) 3 To become familiar with some of the vocabulary terms consider the following: 1. The word constant means “unchanging.” What do you think a constant in mathematics refers to? 2. The word equation looks like the word equal, which means “having the same value.” How do you think this meaning applies to an equation? 3. The word inequality begins with the prefix ‘in’ which means “not”, and has the same root as the word equation. Together, what do you think the prefix and root mean? 4. The word ‘vary’ which is the root of variable means “to change.” How do you think this applies to mathematics? 15-04-2018 In Questions 57 to 67, express each of the given statements as an equation.

13 subtracted from twice of a number gives 3 .

One-fifth of a number is 5 less than that number.

A number is 7 more than one-third of itself.

Six times a number is 10 more than the number.

If 10 is subtracted from half of a number, the result is 4.

Subtracting 5 from p, the result is 2.

Five times a number increased by 7 is 27.

Mohan is 3 years older than Sohan. The sum of their ages is 43 years.

If 1 is subtracted from a number and the difference is multiplied by , the result is 7.

A number divided by 2 and then increased by 5 is 9.

The sum of twice a number and 4 is 18.

The age of Sohan Lal is four times that of his son Amit. If the difference of their ages is 27 years, find the age of Amit.

A number exceeds the other number by 12. If their sum is 72, find the numbers.

Seven times a number is 12 less than thirteen times the same number. Find the number.

The interest received by Karim is 30 more than that of Ramesh. If the total interest received by them is 70, find the interest received by Ramesh.

Subramaniam and Naidu donate some money in a Relief Fund. The amount paid by Naidu is 125 more than that of Subramaniam. If the total money paid by them is 975, find the amount of money donated by Subramaniam. 15-04-2018

In a school, the number of girls is 50 more than the number of boys. The total number of students is 1070. Find the number of girls.

Two times a number increased by 5 equals 9. Find the number.

9 added to twice a number gives 13. Find the number.

1 subtracted from one-third of a number gives 1. Find the number.

After 25 years, Rama will be 5 times as old as he is now. Find his present age.

After 20 years, Manoj will be 5 times as old as he is now. Find his present age.

My younger sister's age today is 3 times, what it will be 3 years from now minus 3 times what her age was 3 years ago. Find her present age.

If 45 is added to half a number, the result is triple the number. Find the number.

In a family, the consumption of wheat is 4 times that of rice. The total consumption of the two cereals is 80 kg. Find the quantities of rice and wheat consumed in the family.

In a bag, the number of one rupee coins is three times the number of two rupees coins. If the worth of the coins is 120, find the number of 1 rupee coins.

Anamika thought of a number. She multiplied it by 2, added 5 to the product and obtained 17 as the result. What is the number she had thought of ?

One of the two numbers is twice the other. The sum of the numbers is 12. Find the numbers.

The sum of three consecutive integers is 5 more than the smallest of the integers. Find the integers.

A number when divided by 6 gives the quotient 6. What is the number?

The perimeter of a rectangle is 40m. The length of the rectangle is 4 m less than 5 times its breadth. Find the length of the rectangle. 15-04-2018

Each of the 2 equal sides of an isosceles triangle is twice as large as the third side. If the perimeter of the triangle is 30 cm, find the length of each side of the triangle.

The sum of two consecutive multiples of 2 is 18. Find the numbers.

Two complementary angles differ by 20°. Find the angles.

150 has been divided into two parts such that twice the first part is equal to the second part. Find the parts.

In a class of 60 students, the number of girls is one third the number of boys. Find the number of girls and boys in the class.

Two-third of a number is greater than one-third of the number by 3. Find the number.

A number is as much greater than 27 as it is less than 73. Find the number.

A man travelled two fifth of his journey by train, one-third by bus, one-fourth by car and the remaining 3 km on foot. What is the length of his total journey?

Twice a number added to half of itself equals 24. Find the number.

Thrice a number decreased by 5 exceeds twice the number by 1. Find the number.

A girl is 28 years younger than her father. The sum of their ages is 50 years. Find the ages of the girl and her father.

The length of a rectangle is two times its width. The perimeter of the rectangle is 180 cm. Find the dimensions of the rectangle.

Look at this riddle? If she answers the riddle correctly how ever will she pay for the pencils? 15-04-2018

In a certain examination, a total of 3768 students secured first division in the years 2006 and 2007. The number of first division in 2007 exceeded those in 2006 by 34. How many students got first division in 2006?

Radha got 17,480 as her monthly salary and over-time. Her salary exceeds the over-time by 10,000. What is her monthly salary ?

If one side of a square is represented by 18x – 20 and the adjacent side is represented by 42 – 13x, find the length of the side of the square.

Follow the directions and correct the given incorrect equation, written in Roman numerals:

• (a)Remove two of these matchsticks to make a valid equation:
• (b)Move one matchstick to make the equation valid. Find two different solutions.

What does a duck do when it flies upside down? The answer to this riddle is hidden in the equation given below: If i + 69 = 70, then i = ? If 8u = 6u + 8, then u =? If 4a = –5a + 45, then a = ? if 4q + 5 = 17, then q =? If –5t – 60 = – 70, then t = ? If s + 98 = 100, then s =? If p + 9 = 24, then p =_____? If 3c = c +12, then c =_____? If 3 (k + 1) = 24, then k =_____? For riddle answer : substitute the number for the letter it equals 1 2 3 4 5 6 7 8 4 9 15-04-2018

The three scales below are perfectly balanced if • = 3. What are the values of ∆ and * ?

• a.
• b.c.

The given figure represents a weighing balance. The weights of some objects in the balance are given. Find the weight of each square and the circle. 1. Crossword Number Puzzle Get cracking on the following questions to fill the crossword puzzle as per mentioned clues of down and across. Clue number is written at the corner of boxes. Answers of Clue have to the filled up in their respective boxes. Down 1: I spent one third of my sleeping time while dreaming. If I dreamt for 3 hours, then how long did I sleep? Down 2: I ran around three sides of a square park whose perimeter is 200 m. How far am I from the starting point? Down 4: I purchased three sarees and was left with 1000 out of my savings of 10000. How much is each saree worth? Down 8: I have 4 coins worth 50 paise each and a few coins of 1 each. If I have 45 in total, how many coins of 1 do I possess? 15-04-2018 Down 9: The unequal angle of an isosceles triangle measures 120. How much is each of the remaining angles? Down 11: For what value of y is 3 (y – 1) + 7 = 40? Across 3: Out of 40 chocolates, Ram and I shared in the ratio 1 : 3. How many chocolates did Ram get? Across 5: Sum of two consecutive numbers is 111. What is the smaller number of the two? Across 7: Out of a flock of birds, half flew away while one got injured, if 244 remain, then how many did we begin with? Across 10: If 2x + 7 = 1573, then x = ? Across 12: What value of z satisfies z + 8 = 58 ? 2. Crossword Puzzle Fill the following crossword puzzle as per the mentioned clues of down and across. Clue’s number is written at the corner of boxes. Answers of Clues have to fill up in their respective boxes. Down 1: A mathematical statement with two expressions that have same value. Down 2: The property that states a(b + c) = ab + ac 15-04-2018 Down 5: An operation that undoes another operation. Across 3: The expression which can be formed by performing mathematical operations on variables and constants. Across 4: A number that does not change. Across 6: A number that multiply the variable. Across 7: A letter that represents an unknown number. 15-04-2018 3.Game Time There are nine identical looking pearl. Eight are real and one is fake. Using a balance scale that consists of two pans, you must find the fake pearl. The real pearls weigh the same and the fake weighs less. Also, the scale can be used maximum twice. Now find the Phony! [Hint: Divide the pearls into three equal groups and then proceed for weighing.] 15-04-2018