Class 7 Mathematics · 134 questions · 124 with answers
Madhre is standing in the middle of a bridge which is 20 m above the water level of a river. If a 35 m deep river is flowing under the bridge (see Fig. 1.1), then the vertical distance between the foot of Madhre and bottom level of the river is:
Solution: The correct answer is (a) [Vertical distance = 20 m + 35 m = 55 m]
[(– 10) × (+ 9)] + ( – 10) is equal to
Solution: Correct answer is (b)
–16 ÷ [8 ÷ (–2)] is equal to
Solution: Correct answer is (c), In Examples 4 and 5, fill in the blanks to make the statements true.
(– 25) × 30 = – 30 × _______.
Solution: 25
75 ÷ _______ = – 75
Solution: –1 If the signs are the same If the signs are different Find the sum of the values of Find the difference of the values of integers without sign, and then integers without sign (subtract use the same sign as the integers lower value integer from greater have. value integer) and then use the sign of the integer with the greater value. In Examples 6 and 7, state whether the statements are True or False.
(–5) × (–7) is same as (–7) × (–5)
Solution : True
(– 80) ÷ (4) is not same as 80 ÷ (–4)
Solution: False
Find the odd one out* of the four options in the following:
Solution: Here – 2 × 24 = – 48, – 4×12 = – 48 and – 6×8 = – 48 * To find odd one out, you have to look for a pattern between the numbers and then find out which option is not on that pattern. 15-04-2018 All the pairs i.e. (–2, 24); (– 4, 12); (– 6, 8) give same answer on multiplication, whereas –3 × 10 = –30, gives a different answer. So, odd one is (b).
Find the odd one out of the four options given below:
Solution: Here –3 + (–6) = – 9, +1 + (–10) = –9 and – 2 + (–7) = –9 All the above pairs i.e. (– 3, – 6); (+1, –10); (–2, –7) give same answer on adding, whereas – 4 + (– 9) = –13, gives a different answer. So, odd one out is (d).
Match the integer in Column I to an integer in Column II so that the sum is between –11 and – 4 Column I Column II (a) – 6
Solution: (a) ↔ (iii) because –6 + (+1) = –5, which lies between –11 and –4. (b) ↔ (i) because +1 + (–11) = –10 which lies between –11 and –4. (c) ↔ (iv) because +7 + (–13) = –6 which lies between –11 and –4 (d) ↔ (ii) because –2 + (–5) = –7 which lies between –11 and –4.
If a is an integer other than 1 and –1, match the following: Column I Column II (a) a ÷ (–1)
Solution: (a) ↔ (iv) (b) ↔ (iii) (c) ↔ (ii) (d) ↔ (i)
Write a pair of integers whose sum is zero (0) but difference is 10. 15-04-2018
Solution: Since sum of two integers is zero, one integer is the additive inverse of other integer, like – 3, 3; – 4, 4 etc. But the difference has to be 10. So, the integers are 5 and – 5 as 5–(–5) is 10. Words Numbers Formula To subtract an integer, 3 – 7 = 3 + (–7) a – b = a + (–b) add its inverse 5 – (–8) = 5 + 8 a – (– b) = a + b
Write two integers which are smaller than –3, but their difference is greater than –3.
Solution: – 5 and – 4 are smaller than – 3 but their difference is (–4) – (–5) = 1 which is greater than – 3. – 6 and –10 are smaller than – 3 but their difference is (–6) – (–10) = 4 which is greater than – 3.
Write a pair of integers whose product is – 15 and whose difference is 8.
Solution: There are few pairs of integers whose product is – 15. e.g. – 1 × 15 –3× 5
If ∆ is an operation such that for integers a and b we have a ∆ b = a × a + b × b – a × b, then find (–3) ∆ 2.
Solution : –3 ∆ 2 = (–3) × (–3) + 2 × 2 – (–3) × 2 = 9 + 4 – ( – 6) = 13 + 6 = 19.
In an objective type test containing 25 questions. A student is to be awarded +5 marks for every correct answer, –5 for every incorrect answer and zero for not writing any answer. Mention the ways of scoring 110 marks by a student.
Solution: Marks scored = +110 15-04-2018 So, minimum correct responses = 110 ÷ (+5) = 22 Case 1 Correct responses = 22 Marks for 1 correct response = + 5 Marks for 22 correct respones = +110 (22 × 5) Marks scored = +110 Marks obtained for incorrect answer = 0 So, no incorrect response And, therefore, 3 were unattempted Case 2 Correct responses = 23 Marks from 23 correct responses = + 115 (23 × 5) Marks scored = + 110 Marks obtained for incorrect answers = 110 – (+115) = –5 Marks for 1 incorrect answer = –5 Number of incorrect responses = (–5) ÷ (–5) =1 So, 23 correct, 1 incorrect and 1 unattempted. Case 3 Correct responses = 24 Marks from 24 correct responses = + 120 (24 × 5) Marks scored = + 110 Marks obtained for incorrect answers = +110 – (+120) = –10 Number of incorrect responses = (–10) ÷ (–5) =2 Thus the number of questions = 24 + 2 = 26. Whereas, total number of questions is 25. So, this case is not possible. So, the possible ways are: • 22 correct, 0 incorrect, 3 unattempted • 23 correct, 1 incorrect, 1unattempted. If the signs are the same, the sign of the answer is positive. If the signs are different, the sign of the answer is negative. 15-04-2018
A boy standing on the third stair on a staircase goes up by five more stairs. Which stair is he standing at now? At which step will he be after he comes down by 2 stairs?
Solution: Understand and Explore the Problem • What do you know? The current position of the boy. • What are we trying to find? The new position of the boy after he takes five more stairs in the same direction. Plan a Strategy • Going up one stairs is equal to + 1 and coming down by one stairs is equal to – 1. Ground level is taken as 0. Solve • He is currently at the third stair i.e. at (+3). • He goes up 5 stairs in the same direction. • Since 3 + 5 = 8. Therefore, he is at 8th stair on the staircase. Now, the boy comes down by 2 stairs. Since he comes down in opposite direction i.e. downwards by 2 stairs (i.e. –2), so 8 + (–2) = 8 – 2 = 6. He is at 6th step now. Revise Since the boy moves 5 stairs up and then 2 stairs down, therefore, as a result he moves (5 –2) stairs i.e. 3 stairs up. As he was on stair 3 and since he goes up by 3 stairs (i.e. +3), so, he is at (3 + 3) the i.e 6th step now. 1. Can you find the position of the boy if he comes down further by 3 more stairs? 15-04-2018 In the Questions 1 to 25, there are four options, out of which only one is correct. Write the correct one. 1. When the integers 10, 0, 5, – 5, – 7 are arranged in descending or ascending order, them find out which of the following integers always remains in the middle of the arrangement. (a) 0 (b) 5 (c) – 7 (d) – 5 2. By observing the number line (Fig. 1.2), state which of the following statements is not true. Fig. 1.2 (a) B is greater than –10 (b) A is greater than 0 (c) B is greater than A (d) B is smaller than 0 3. By observing the above number line (Fig. 1.2), state which of the following statements is true. (a) B is 2 (b) A is – 4 (c) B is –13 (d) B is – 4 1. Compare the sums 10 + (–22) and –10 + 22. 2. Describe how to add the following addition expressions on a number line, 9 + (– 13) and –13 + 9. Then compare the sums.
× (– 5) 15 × (– 1) but difference of –3 and 5 or –5 and 3 is 8. So the required pair of integers is – 3, 5 and – 5, 3.
(d)
Next three consecutive numbers in the pattern 11, 8, 5, 2, --, --, --
(d) – 1, – 4, – 7 15-04-2018
The next number in the pattern – 62, – 37, – 12 _________ is
(b) 13
Which of the following statements is not true?
(c) When a positive integer and a negative integer is added we always get a negative integer.
On the following number line value ‘Zero’ is shown by the point
(c) Z
If ⊗, O, and • represent some integers on number line, then descending order of these numbers is
(c) , , ⊗, •
On the number line, the value of (–3) × 3 lies on right hand side of
(a) – 10
The value of 5 ÷ (–1) does not lie between
(b) 0 and 10
Water level in a well was 20m below ground level. During rainy season, rain water collected in different water tanks was drained 15-04-2018 into the well and the water level rises 5 m above the previous level. The wall of the well is 1m 20 cm high and a pulley is fixed at a height of 80 cm. Raghu wants to draw water from the well. The minimum length of the rope that he can use is
This question refers to a figure in the original PDF.
(a) 17 m
(– 11) × 7 is not equal to
(c) (– 11) × (– 7)
(– 10) × (– 5) + (– 7) is equal to
(d) 43
Which of the folllowing is not the additive inverse of a ?
(a) – (– a)
Which of the following is the multiplicative identity for an integer a ?
(b) 1
[(– 8) × ( – 3)] × (– 4) is not equal to
(d) ( – 8) × (– 3) – (– 8) × (– 4)
(– 25) × [6 + 4] is not same as
(c) (– 25) × 6 × 4
– 35 × 107 is not same as
(c) – 35 × 7 + 100
( – 43) × ( – 99) + 43 is equal to
(a) 4300
(– 16) ÷ 4 is not same as
(a) ( – 4) ÷ 16
Which of the following does not represent an integer?
(d) (– 12) ÷ 5
Which of the following is different from the others?
(c) (– 5) × ( –1)
Which of the following shows the maximum rise in temperature?
(b) – 10° to + 1°
If a and b are two integers, then which of the following may not be an integer?
(d) a ÷ b
For a non-zero integer a which of the following is not defined?
(a) a ÷ 0
(c) (–6, 1)
(d) (–9, +7) 15-04-2018
(c) (–9) × (–5) × (–6) × 3
(d) (–32) ÷ 9
(b) (–1) × (–1) × (–1)
(–a) + b = b + Additive inverse of __________.
a
________ ÷ (–10) = 0
0
(–157) × (–19) + 157 = ___________
3140
[(–8) + ______ ] + ________ = ________ + [(–3) + ________ ] = –3
–3, 8, (–8), 8
On the following number line, (–4) × 3 is represented by the point _________.
D
If x, y and z are integers then (x+___ ) + z = _____ + (y + _____ )
y, x, z
(– 43) + _____ = – 43
0
(– 8) + (– 8) + (– 8) = _____ × (– 8)
3
11 × (– 5) = – ( _____ × _____ ) = _____
11, 5, –55
(– 9) × 20 = _____
–180
(– 23) × (42) = (– 42) × _____
23
While multiplying a positive integer and a negative integer, we multiply them as ________ numbers and put a ________ sign before the product.
Whole, Negative
If we multiply ________ number of negative integers, then the resulting integer is positive.
Even
If we multiply six negative integers and six positive integers, then the resulting integer is _______. 15-04-2018
Positive
If we multiply five positive integers and one negative integer, then the resulting integer is _______.
Negative
is the multiplicative identity for integers.
1
We get additive inverse of an integer a when we multiply it by _________.
(–1)
( – 25) × ( – 2) =
50
(– 5) × ( – 6) × ( – 7) =
–210
3 × ( – 1 ) × ( – 15) =
45
[12 × ( – 7)] × 5 = × [(– 7) × ]
12, 5
23 × ( – 99) = × ( – 100 + ) = 23 × + 23 ×
23, 1, –100, 1
× ( – 1) = – 35
35
× ( – 1) = 47 1. Explain why 10 – (– 10) does not equal – 10 – 10. 2. Describe the answer that you get when you subtract a greater number from a lesser number.
–47
88 × = – 88
–1
× (–93) = 93
–1
( – 40) × = 80
–2
× (–23) = – 920
40
When we divide a negative integer by a positive integer, we divide them as whole numbers and put a sign before quotient.
Minus
When –16 is divided by _________ the quotient is 4. 15-04-2018
Negative integer
Division is the inverse operation of ____________
Multiplication
65 ÷ ( – 13) =
–5
( – 100) ÷ ( – 10) =
10
( – 225) ÷ 5 =
– 45
_____÷ ( – 1 ) = – 83
83
_____ ÷ ( – 1) = 75
–75
51 ÷ _____ = – 51
–1
113 ÷ _____ = – 1
–113
(– 95) ÷ _____ = 95
–1
( – 69) ÷ ( 69) = _____
– 1
( – 28) ÷ ( – 28) = _____ In Questions 72 to 108, state whether the statements are True or False.
1
5 – ( – 8) is same as 5 + 8.
True
(– 9) + (– 11) is greater than (– 9) – ( – 11).
False
Sum of two negative integers always gives a number smaller than both the integers.
True
Difference of two negative integers cannot be a positive integer.
False
We can write a pair of integers whose sum is not an integer.
False
Integers are closed under subtraction.
True
(– 23) + 47 is same as 47 + (– 23).
True
When we change the order of integers, their sum remains the same.
True
When we change the order of integers their difference remains the same.
False
Going 500 m towards east first and then 200 m back is same as going 200 m towards west first and then going 500 m back.
True
(– 5) × (33) = 5 × ( – 33)
True
(– 19) × (– 11) = 19 × 11 15-04-2018
True
(– 20) × ( 5 – 3) = (– 20) × ( – 2)
False
4 × ( – 5) = ( – 10) × ( – 2)
False
( – 1) × ( – 2) × ( – 3) = 1 × 2 × 3
False
– 3 × 3 = – 12 – ( – 3)
True
Product of two negative integers is a negative integer.
False
Product of three negative integers is a negative integer.
True
Product of a negative integer and a positive integer is a positive integer.
False
When we multiply two integers their product is always greater than both the integers.
False
Integers are closed under multiplication.
True
(– 237) × 0 is same as 0 × (– 39)
True
Multiplication is not commutative for integers.
False
(–1) is not a multiplicative identity of integers.
True
99 × 101 can be written as (100 – 1) × (100 + 1)
True
If a, b, c are integers and b ≠ 0 then, a× (b – c) = a × b – a × c
False
(a + b) × c = a × c + a × b
True
a × b = b × a
True
a ÷ b = b ÷ a
False
a – b = b – a 1. List all possible multiplication and division statements for the integers with 5, –5, 6, – 6 and 30, –30. For example, 5 × 6 = 30. 2. Compare the sign of the product of two negative integers with the sign of the sum of two negative integers. 3. Suppose the product of two integers is positive. What do you know about the signs of the integers? 15-04-2018
False
a ÷ (–b) = – (a ÷ b)
True
a ÷ ( –1) = – a
True
Multiplication fact (–8) × (–10) = 80 is same as division fact 80 ÷ (– 8) = (–10)
True
Integers are closed under division.
False
[(–32) ÷ 8 ] ÷ 2 = –32 ÷ [ 8 ÷ 2]
False
The sum of an integer and its additive inverse is zero (0).
True
The successor of 0 × (–25) is 1 × (–25)
False
Observe the following patterns and fill in the blanks to make the statements true:
Science Application: An atom consists of charged particles called electrons and protons. Each proton has a charge of +1 and each electron has a charge of –1. Remember number of electrons is equal to number of protons, while answering these questions:
An atom changes to a charged particle called ion if it loses or gains electrons. The charge on an ion is the charge on electrons plus charge 15-04-2018 on protons. Now, write the missing information in the table given below: Name of Ion Proton Charge Electron Charge Ion Charge Hydroxide ion +9 — –1 Sodium ion +11 — +1 Aluminium ion +13 –10 — Oxide ion +8 –10 — Plan a Strategy • Some problems contain a lot of information. Read the entire problem carefully to be sure you understand all of the facts. You may need to read it over several times perhaps aloud so that you can hear yourself. • Then decide which information is the most important (prioritise). Is there any information that is absolutely necessary to solve the problem? This information is the most important. • Finally, put the information in order (sequence). Use comparison words like before, after, longer, shorter, and so on to help you. Write down the sequence before you try to solve the problem. Read the problem given below and then answer the questions that follow: • Five friends are standing in a line waiting for the opening of a show. They are in line according to their arrival. Shreya arrived 3 minutes after Sachin. Roy took his place in line at 9:01 P.M. He was 1 minute behind Reena and 7 minutes ahead of Shreya. The first person arrived at 9:00 P.M. Babu showed up 6 minutes after the first person. List the time of each person’s arrival.
Social Studies Application: Remembering that 1AD came immediately after 1BC, while solving these problems take 1BC as –1 and 1AD as +1.
The table shows the lowest recorded temperatures for each continent. Write the continents in order from the lowest recorded temperature to the highest recorded temperature. The Lowest Recorded Temperatures Continent Temperature (in Fahrenheit) Africa –110 Antarctica –1290 Asia –900 Australia – 90 Europe – 67 0 North America – 810 South America –270
Antarctica, Asia, N. America, Europe, S. America, Africa, Australia.
Write a pair of integers whose product is –12 and there lies seven integers between them (excluding the given integers).
–2, 6
From given integers in Column I match an integer of Column II so that their product lies between –19 and –6: Column I Column II –5 1 6 –1 –7 3 8 –2 15-04-2018
–5 → 3, 6 → –2, –7 → 1, 8 → –1,
Write a pair of integers whose product is – 36 and whose difference is 15.
–3, 12
Match the following Column I Column II (a) a ×1
(i) Additive inverse of a (b) 1
(ii) Additive identity (c) ( – a) ÷ ( – b)
(iii) Multiplicative identity (d) a × ( – 1)
(iv) a ÷ ( – b) (e) a ×0 (v) a÷b (f) ( –a) ÷ b (vi) a (g) 0 (vii) – a (h) a ÷ (–a) (viii) 0 (i) –a (ix) –1
Hint: 500 + 200 + 150 –120 –240
You have 500 in your savings account at the beginning of the month. The record below shows all of your transactions during the month. How much money is in your account after these transactions? Cheque No. Date Transaction Payment Deposit Description 384102 4/9 Jal Board 120 200 275146 12/9 Deposit 384103 22/9 LIC India 240 150 801351 29/9 Deposit Is it not true? If + is a friend and – is an enemy. +×+=+ 1. Your friend’s friend is your friend. +×–=– 2. Your friend’s enemy is your enemy. 3. Your enemy’s friend is your enemy. –×+=– –×–=+ 4. Your enemy’s enemy is your friend. 15-04-2018
(a) Write a positive integer and a negative integer whose sum is a negative integer.
(b) Write a positive integer and a negative integer whose sum is a positive integer.
(c) Write a positive integer and a negative integer whose difference is a negative integer.
(d) Write a positive integer and a negative integer whose difference is a positive integer. (e) Write two integers which are smaller than – 5 but their difference is – 5. (f) Write two integers which are greater than – 10 but their sum is smaller than – 10. (g) Write two integers which are greater than – 4 but their difference is smaller than – 4. (h) Write two integers which are smaller than – 6 but their difference is greater than – 6. (i) Write two negative integers whose difference is 7. (j) Write two integers such that one is smaller than –11, and other is greater than –11 but their difference is –11. (k) Write two integers whose product is smaller than both the integers. (l) Write two integers whose product is greater than both the integers.
What’s the Error? Ramu evaluated the expression –7 – (–3) and came up with the answer –10. What did Ramu do wrong?
Ramu went wrong in solving –(–3) and took it as –3 only.
What’s the Error? Reeta evaluated – 4 + d for d = – 6 and gave an answer of 2. What might Reeta have done wrong?
Reeta went wrong in sloving + (–6) and took it as +6. 122.(a) C (b) D (c) A, C, B, D 123. 356 m.
The table given below shows the elevations relative to sea level of four locations. Taking sea level as zero, answer the following questions: Location Elevation (in m) A –180 B 1600 C – 55 D 3200 15-04-2018
You are at an elevation 380 m above sea level as you start a motor ride. During the ride, your elevation changes by the following metres: 540 m, –268 m, 116 m, –152 m, 490 m, –844 m, 94 m. What is your elevation relative to the sea level at the end of the ride?
Evaluate the following, using distributive property.
(i) – 39 × 99
(ii) (– 85) × 43 + 43 × ( – 15)
(iii) 53 × ( – 9) – ( – 109) × 53
(iv) 68 × (–17) + ( –68) × 3
If * is an operation such that for integers a and b we have a * b = a × b + (a × a + b × b) then find
(i) ( – 3) * (– 5)
(ii) ( – 6) * 2
If ∆ is an operation such that for integers a and b we have a ∆ b = a × b – 2 × a × b + b × b (–a) × b + b × b then find
(i) 4 ∆ ( – 3)
(ii) ( – 7) ∆ ( – 1) Also show that 4 ∆ ( – 3) ≠ (– 3) ∆ 4 and ( – 7) ∆ ( – 1) ≠ ( – 1) ∆ (– 7)
Below u, v, w and x represent different integers, where u = –4 and x ≠ 1. By using following equations, find each of the values: u×v=u x×w=w u+x=w
(a) v
(b) w
(c) x Explain your reasoning using the properties of integers.
Height of a place A is 1800 m above sea level. Another place B is 700 m below sea level. What is the difference between the levels of these two places?
2500m
The given table shows the freezing points in 0F of different gases at sea level. Convert each of these into 0C to the nearest integral value using the relation and complete the table, C = ( F − 32) 15-04-2018 Gas Freezing Point at Freezing Point at Sea Level (°F) Sea Level (°C) Hydrogen –435 Krypton –251 Oxygen –369 Helium –458 Argon –309
Hydrogen –2590C, Krypton –1570C, Oxygen –2230C, Helium –2720C, Argon –1890C
Sana and Fatima participated in an apple race. The race was conducted in 6 parts. In the first part, Sana won by 10 seconds. In the second part she lost by 1 minute, then won by 20 seconds in the third part and lost by 25 seconds in the fourth part, she lost by 37 seconds in the fifth part and won by 12 seconds in the last part. Who won the race finally?
Fatima.
A green grocer had a profit of 47 on Monday, a loss of 12 on Tuesday and loss of 8 on Wednesday. Find his net profit or loss in 3 days.
Net profit 27
In a test, +3 marks are given for every correct answer and –1 mark are given for every incorrect answer. Sona attempted all the questions and scored +20 marks though she got 10 correct answers.
(i) How many incorrect answers has she attempted?
(ii) How many questions were given in the test?
In a true-false test containing 50 questions, a student is to be awarded 2 marks for every correct answer and –2 for every incorrect answer and 0 for not supplying any answer. If Yash secured 94 marks in a test, what are the possibilities of his marking correct or wrong answer?
Since Yash scored 94 marks So, Minimum correct responses = 94 ÷ (+2) = 47, Two possitities are there:
A multistorey building has 25 floors above the ground level each of height 5m. It also has 3 floors in the basement each of height 5m. A lift in building moves at a rate of 1m/s. If a man starts from 50m above the ground, how long will it take him to reach at 2nd floor of basement?
Taking today as zero on the number line, if the day before yesterday is 17 January, what is the date 3 days after tomorrow? Explain how integers are used in real life to manage a bank account. Explain whether –1, – 4, and 5 are additive inverses. 15-04-2018
The highest point measured above sea level is the summit of Mt. Everest which is 8,848m above sea level and the lowest point is challenger Deep at the bottom of Mariana Trench which is 10911m below sea level. What is the vertical distance between these two points? Puzzle 1 Fill in the blank space of the following magic square so that the sum of the numbers in each row, each column and each of the diagonals is – 6. (i) –1 3 –2 (ii) In this magic square, sum of the numbers in every row, column and each of the diagonals is – 2. Fill in the blank: 7 –6 0 –2 –5 6 –8 Puzzle 2 If a * b means a × b + 2 and a # b means – a + b –(–3), then find the value of the following: (i) – 4 * 3 (ii) (–3) * (–2) (iii) (–7) # (–3) (iv) 2 # (–4) (v) 7 * (–5) (vi) (–7 * 2) # 3 Next, match these answers with suitable letters by looking at the table below and arrange them in increasing order of integers to decode the name of the mathematician: 15-04-2018 Integers –9 14 –3 4 –10 8 –33 –21 7 18 Letters P Y C T U I E G L D Puzzle 3 ‘Equinoxes’ are the two days of the year when the sun is directly above the earth’s equator, due to which the days and nights are of nearly equal length everywhere on the earth. Find the name of the month of autumn equinox using suitable properties of integers by solving the following questions. Match your answer with the letter given in the table and fill it in the box provided in each question.
This question refers to a figure in the original PDF.