109. Observe the following patterns and fill in the blanks to make the statements true:
(a)
−5 × 4 = −20
−5 × 3 = −15 = −20 − (−5)
−5 × 2 = ______ = −15 − (−5)
−5 × 1 = ______ = ______
−5 × 0 = 0 = ______
−5 × −1 = 5 = ______
−5 × −2 = ______ = ______
(b)
7 × 4 = 28
7 × 3 = ______ = 28 − 7
7 × 2 = ______ = ______ − 7
7 × 1 = 7 = ______ − 7
7 × 0 = ______ = ______ − ______
7 × −1 = −7 = ______ − ______
7 × −2 = ______ = ______ − ______
7 × −3 = ______ = ______ − ______
(a) −5 × 2 = −10 = −15 − (−5)
−5 × 1 = −5 = −10 − (−5)
−5 × 0 = 0 = −5 − (−5)
−5 × −1 = 5 = 0 − (−5)
−5 × −2 = 10 = 5 − (−5)
(b) 7 × 3 = 21 = 28 − 7
7 × 2 = 14 = 21 − 7
7 × 1 = 7 = 14 − 7
7 × 0 = 0 = 7 − 7
7 × −1 = −7 = 0 − 7
7 × −2 = −14 = −7 − 7
7 × −3 = −21 = −14 − 7
110. Science Application: An atom consists of electrons and protons. Each proton has charge +1 and each electron has charge −1. (Number of electrons = number of protons in a neutral atom.)
(a) What is the charge on an atom?
(b) What will be the charge on an atom if it loses an electron?
(c) What will be the charge on an atom if it gains an electron?
(a) 0
(b) +1
(c) −1
111. An atom becomes an ion if it loses or gains electrons. The ion charge is (sum of proton charges) + (sum of electron charges). Complete the table.
| 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 | — |
| Name of Ion | Proton Charge | Electron Charge | Ion Charge |
|---|---|---|---|
| Hydroxide ion | +9 | −10 | −1 |
| Sodium ion | +11 | −10 | +1 |
| Aluminium ion | +13 | −10 | +3 |
| Oxide ion | +8 | −10 | −2 |
112. Social Studies Application:
(a) The Greco–Roman era in Egypt started in 330 BC and ended in 395 AD. How long did it last?
(b) Bhaskaracharya was born in 1114 AD and died in 1185 AD. What was his age at death?
(c) Turks ruled Egypt in 1517 AD. Queen Nefertis ruled Egypt about 2900 years before the Turks. In which year did she rule?
(d) Greek mathematician Archimedes lived between 287 BC and 212 BC and Aristotle lived between 380 BC and 322 BC. Who lived during an earlier period?
(a) 725 years
(b) 71 years
(c) 1383 BC
(d) Aristotle
113. 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.
| Continent | Temperature (in °F) |
|---|---|
| Africa | −11° |
| Antarctica | −129° |
| Asia | −90° |
| Australia | −9° |
| Europe | −67° |
| North America | −81° |
| South America | −27° |
Antarctica, Asia, North America, Europe, South America, Africa, Australia
Write a pair of integers whose product is −12 and there lie seven integers between them (excluding the given integers).
−2 and 6
115. From the 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 |
−5 → 3, 6 → −2, −7 → 1, 8 → −1
Write a pair of integers whose product is −36 and whose difference is 15.
−3 and 12
117. Match the following
| Column I | Column II |
|---|---|
| (a) \(a\times1\) | (i) Additive inverse of \(a\) |
| (b) 1 | (ii) Additive identity |
| (c) \((−a)÷(−b)\) | (iii) Multiplicative identity |
| (d) \(a\times(−1)\) | (iv) \(a÷(−b)\) |
| (e) \(a\times0\) | (v) \(a÷b\) |
| (f) \((−a)÷b\) | (vi) \(a\) |
| (g) 0 | (vii) \(−a\) |
| (h) \(a÷(−a)\) | (viii) 0 |
| (i) \(−a\) | (ix) −1 |
(a) → (vi), (b) → (iii), (c) → (v), (d) → (vii), (e) → (viii), (f) → (iv), (g) → (ii), (h) → (ix), (i) → (i)
118. You have ₹500 in your savings account at the beginning of the month. The record below shows all transactions during the month. How much money is in your account after these transactions?
| Cheque No. | Date | Transaction Description | Payment | Deposit |
|---|---|---|---|---|
| 384102 | 4/9 | Jal Board | ₹ 120 | ₹ 200 |
| 275146 | 12/9 | Deposit | — | — |
| 384103 | 22/9 | LIC India | ₹ 240 | ₹ 150 |
| 801351 | 29/9 | Deposit | — | — |
₹ 490
119. Give one example for each (many answers possible):
(a) A positive and a negative integer whose sum is a negative integer.
(b) A positive and a negative integer whose sum is a positive integer.
(c) A positive and a negative integer whose difference is a negative integer.
(d) A positive and a negative integer whose difference is a positive integer.
(e) Two integers smaller than −5 but their difference is −5.
(f) Two integers greater than −10 but their sum is smaller than −10.
(g) Two integers greater than −4 but their difference is smaller than −4.
(h) Two integers smaller than −6 but their difference is greater than −6.
(i) Two negative integers whose difference is 7.
(j) Two integers such that one is smaller than −11 and the other is greater than −11 but their difference is −11.
(k) Two integers whose product is smaller than both the integers.
(l) Two integers whose product is greater than both the integers.
(a) 4 + (−6) = −2
(b) 8 + (−2) = 6
(c) −7 − 2 = −9
(d) 4 − (−3) = 7
(e) −12 − (−7) = −5
(f) −4 + (−7) = −11 < −10
(g) −1 − 4 = −5 < −4
(h) −8 − (−9) = 1 > −6
(i) −2 − (−10) = 8
(j) −20 − (−9) = −11
(k) −3 × 5 = −15
(l) 4 × 6 = 24
120. What’s the Error? Ramu evaluated the expression \(−7 − (−3)\) and got \(−10\). What did Ramu do wrong?
He treated \(−(−3)\) as \(−3\) instead of \(+3\).
121. What’s the Error? Reeta evaluated \(−4 + d\) for \(d = −6\) and got \(2\). What did she do wrong?
She took \(+ (−6)\) as \(+6\) instead of \(−6\).
122. The elevations (relative to sea level) of four locations are:
| Location | Elevation (m) |
|---|---|
| A | −180 |
| B | 1600 |
| C | −55 |
| D | 3200 |
(a) Which location is closest to sea level?
(b) Which location is farthest from sea level?
(c) Arrange the locations from the least to the greatest elevation.
(a) C
(b) D
(c) A, C, B, D
You start at 380 m above sea level. During a ride, your elevation changes by: +540 m, −268 m, +116 m, −152 m, +490 m, −844 m, +94 m. What is your final elevation relative to sea level?
356 m
124. Evaluate using the distributive property:
(i) \(−39\times99\)
(ii) \((−85)\times43 + 43\times(−15)\)
(iii) \(53\times(−9) − (−109)\times53\)
(iv) \(68\times(−17) + (−68)\times3\)
(i) −3861
(ii) −4300
(iii) 5300
(iv) −1360
125. If * is defined for integers \(a,b\) by
\(a*b=a\times b + (a\times a + b\times b)\), find:
(i) \((−3)* (−5)\)
(ii) \((−6)*2\)
(i) 49
(ii) 28
126. If \(\Delta\) is defined for integers \(a,b\) by \(a\,\Delta\,b = a\times b − 2ab + b^2\):
(i) Compute \(4\,\Delta\,(−3)\) and \((−3)\,\Delta\,4\). Are they equal?
(ii) Compute \((−7)\,\Delta\,(−1)\) and \((−1)\,\Delta\,(−7)\). Are they equal?
(i) 4 Δ (−3) = 21, (−3) Δ 4 = 28, No
(ii) (−7) Δ (−1) = −6, (−1) Δ (−7) = 42, No
127. Let \(u,v,w,x\) be integers with \(u=-4\) and \(x\neq1\). Given
\(u\times v = u\), \(x\times w = w\), \(u + x = w\).
Find (a) \(v\), (b) \(w\), (c) \(x\).
(a) 1
(b) 0
(c) 4
A place A is 1800 m above sea level and place B is 700 m below sea level. What is the difference between their levels?
2500 m
129. Convert the freezing points (°F) to the nearest integral values in °C using \(C=\tfrac{5}{9}(F−32)\) and complete the table.
| Gas | Freezing Point (°F) | Freezing Point (°C) |
|---|---|---|
| Hydrogen | −435 | — |
| Krypton | −251 | — |
| Oxygen | −369 | — |
| Helium | −458 | — |
| Argon | −309 | — |
| Gas | Freezing Point (°F) | Freezing Point (°C) |
|---|---|---|
| Hydrogen | −435 | −259° C |
| Krypton | −251 | −157° C |
| Oxygen | −369 | −223° C |
| Helium | −458 | −272° C |
| Argon | −309 | −189° C |
Sana and Fatima ran an apple race with 6 parts: Sana +10 s, −1 min, +20 s, −25 s, −37 s, +12 s. Who finally won?
Fatima
A green grocer had a profit of ₹47 on Monday, a loss of ₹12 on Tuesday and a loss of ₹8 on Wednesday. Find his net profit or loss in 3 days.
Net profit ₹ 27
132. In a test, +3 marks for every correct answer and −1 for every incorrect answer; 0 for unattempted. Sona attempted all questions and scored +20 though she got 10 correct.
(i) How many incorrect answers did she attempt?
(ii) How many questions were there in the test?
(i) 10
(ii) 30
133. A true–false test has 50 questions. Award +2 for every correct answer, −2 for every incorrect, and 0 for no answer. If Yash secured 94 marks, what are the possibilities of his correct/wrong/unattempted counts?
Minimum correct = 47. Two possible distributions:
1) Correct 47, Unattempted 3.
2) Correct 48, Unattempted 1, Wrong 1.
A multistorey building has 25 floors above ground (each 5 m) and 3 basement floors (each 5 m). A lift moves at 1 m/s. If a man starts from 50 m above ground, how long to reach the 2nd basement floor?
60 seconds (1 minute)
Taking today as zero on the number line, if the day before yesterday is 17 January, what is the date 3 days after tomorrow?
23rd January
(Problem on combined distances/elevations as given in the text.)
19,759 m