CBSE Class 6 Mathematics The Other Side of Zero Worksheet Set 01

Long Answer Questions

Question 1. Reena enters Bela’s Building of Fun at the Welcome Hall, which is Floor 0. In the lift she first presses +2, then −5, and finally +1.
(i) Write the floor she reaches after each press as an expression of the form “Starting Floor + Movement = Target Floor”.
(ii) Which floor does she end on? Is it above or below the ground?
(iii) Which single button press would have taken her straight from the Welcome Hall to that floor?
(iv) Through how many floors in all did the lift move during the three presses?
Answer:
Rule used: Starting Floor + Movement = Target Floor.
(i) First press: 0 + (+2) = +2, the Art Centre.
Second press: (+2) + (−5) = −3, the Cinema Centre.
Third press: (−3) + (+1) = −2, the Video Games shop.
(ii) Reena ends on Floor −2. The number is negative, so the floor is below the ground, two floors under the Welcome Hall.
(iii) Movement needed = Target − Start = (−2) − 0 = −2. One press of −2 would have taken her straight there. Check: (+2) + (−5) + (+1) = −2.
(iv) The lift moved through 2 floors, then 5, then 1: 2 + 5 + 1 = 8 floors in all.
The floors travelled (8) and the final floor (−2) are different things: travel counts every floor crossed, up or down, while the floor number only tells where she stopped.
In simple words: Reena ends on Floor −2, which one press of −2 would do, but the lift actually travelled 8 floors.

Teacher's Note:
1. Do one press at a time and write each step.
2. Distance travelled ignores the signs.

Question 2. The lift attendant uses green (positive) and red (negative) tokens to keep track of his button presses.
(i) The attendant says that putting one more positive token and one more negative token into his pocket does not change the floor he is on. Explain why he is right.
(ii) Show step by step how you would find (+6) + (−9) with tokens. Write how many zero pairs cancel and what is left.
(iii) Write the answer and say which floor of the Building of Fun the attendant is on.
Answer:
(i) A green token stands for +1 and a red token for −1. Together they are worth (+1) + (−1) = 0, a zero pair. Adding 0 leaves any number unchanged, so the value of the pocket, and hence the floor, stays the same.
(ii) Put down 6 green tokens for +6 and 9 red tokens for −9.
Match each green token with a red one: 6 zero pairs are formed and they all cancel.
What is left is 9 − 6 = 3 red tokens, that is −3.
(iii) (+6) + (−9) = −3. The attendant is on Floor −3, the Cinema Centre, three floors below the Welcome Hall.
In simple words: Six zero pairs cancel and three red tokens remain, so he is on Floor −3.

Teacher's Note:
1. The colour of the leftover tokens gives the sign.
2. Zero pairs can always be added or removed.

Question 3. Negative numbers are used in many places outside the classroom.
(i) Write an integer for each of these: a debit of Rs 400; a hill top 900 m above sea level; a temperature of 5 degrees below the freezing point of water; the second floor below the ground of a mall; the year 200 BCE.
(ii) For each of the five situations, name the quantity that is taken as 0 (the reference point).
(iii) Why is it useful to have one agreed reference point in each case?
Answer:
(i) Debit of Rs 400: −400; hill top: +900; temperature: −5 (°C); second floor below the ground: −2; the year 200 BCE: −200.
(ii) Money: a balance of Rs 0 in the account. Height: sea level, 0 m. Temperature: the freezing point of water, 0 °C. Mall: the ground floor, Floor 0. Years: the start of the Common Era, from which BCE years count backwards.
(iii) Without a common starting mark, one person’s ‘above’ could be another person’s ‘below’. Once everyone fixes the same 0, a positive number always means so much on one side and a negative number so much on the other, so everyone can compare and add the numbers in the same way.
In simple words: Every situation needs an agreed zero so that plus and minus mean the same thing to everyone.

Teacher's Note:
1. Name the zero before giving the sign.
2. The same idea works for floors, money, heights, temperatures and years.

Question 4. Five points P, Q, R, S and T are marked on the number line shown below.
(i) Write the integer at each of the five points.
(ii) Arrange these five integers in increasing order.
(iii) How many steps must you walk, and in which direction, to go from Q to S? Write this as a subtraction.
(iv) Write the additive inverse of the integer at P, and state whether it lies to the left or to the right of 0.
Answer:
(i) P = −8, Q = −5, R = −2, S = +4 and T = +7.
(ii) −8 < −5 < −2 < +4 < +7. Numbers grow as we move right, so P (farthest left) is the smallest and T (farthest right) the greatest.
(iii) Movement = Target − Start = (+4) − (−5) = 4 + 5 = +9. So walk 9 steps to the right (the positive direction): 5 steps from −5 to 0 and 4 more from 0 to +4.
(iv) P is −8, and its additive inverse is +8, because (−8) + (+8) = 0. Since +8 is positive, it lies to the right of 0, as far from 0 as P is on the left.
In simple words: Read the points, order them left to right, and note that from Q to S is 9 steps right.

Teacher's Note:
1. Read the number under each dot carefully.
2. A positive movement means walking to the right.

Teacher's Note:
1. Greatest integer = highest point.
2. Above and below distances add up across sea level.

Question 6. Mohan opens a bank account with a balance of Rs 0. In the next four days his passbook shows a credit of Rs 500, a debit of Rs 320, a debit of Rs 400 and a credit of Rs 150, in that order.
(i) Write each of the four entries as an integer.
(ii) Find his balance after each entry.
(iii) After which entry does his balance first become negative, and what does a negative balance mean?
(iv) What is his final balance, and how much must he credit to bring the balance back to Rs 0?
Answer:
(i) Credit of Rs 500: +500; debit of Rs 320: −320; debit of Rs 400: −400; credit of Rs 150: +150.
(ii) After the credit of Rs 500: 0 + (+500) = +500
After the debit of Rs 320: (+500) + (−320) = +180
After the debit of Rs 400: (+180) + (−400) = −220
After the credit of Rs 150: (−220) + (+150) = −70
(iii) The balance first becomes negative after the third entry, the debit of Rs 400, because only Rs 180 was in the account. A balance of −220 means he has taken out Rs 220 more than he put in, so he owes the bank that money.
(iv) The final balance is Rs −70. He must credit Rs 70, since (−70) + (+70) = 0; Rs +70 is the additive inverse of Rs −70.
In simple words: The balance goes 500, 180, −220, −70, so Mohan needs to put in Rs 70.

Teacher's Note:
1. Keep a running total after each entry.
2. To reach 0, add the additive inverse.

Question 7. Subtraction with tokens is possible even when there seem to be too few tokens.
(i) Find (−3) − (−8) with tokens. State how many zero pairs you must put down and what is left.
(ii) Find (−6) − (+4) with tokens in the same way.
(iii) Rewrite both subtractions as additions and check that the answers agree.
(iv) Write, in one sentence, the general rule you have used.
Answer:
(i) Put down 3 red tokens. Eight red tokens must be taken away, but there are only three, so put down 5 zero pairs. Now there are 8 red and 5 green tokens. Take away the 8 red tokens; 5 green tokens are left. Hence (−3) − (−8) = +5.
(ii) Put down 6 red tokens. Four green tokens must be taken away, but there are none, so put down 4 zero pairs, giving 4 green and 10 red tokens. Take away the 4 green tokens; 10 red tokens are left. Hence (−6) − (+4) = −10.
(iii) (−3) − (−8) = (−3) + (+8) = +5 and (−6) − (+4) = (−6) + (−4) = −10. Both agree with the token work.
(iv) Subtracting a number is the same as adding its additive inverse.
In simple words: Add enough zero pairs, remove what you must, and count what is left.

Teacher's Note:
1. The number of zero pairs equals the shortage.
2. Check token answers with the addition rule.

Question 8. Large integers can be added and subtracted on an unmarked number line, which shows only the position of 0.
(i) Using an unmarked number line, evaluate (+150) − (−80).
(ii) Using an unmarked number line, evaluate (−230) + (−70).
(iii) Explain why subtracting a negative number is the same as adding the matching positive number.
(iv) Name the rule of Brahmagupta that says the same thing, and give one more example of it.
Answer:
(i) Mark 0 on a plain line, −80 to its left and +150 to its right. The movement from −80 to +150 is 80 steps to reach 0 and 150 steps more after it.
(+150) − (−80) = 80 + 150 = +230
(ii) Start at −230, to the left of 0, and move 70 steps further left. Both moves are backwards, so they add up:
(−230) + (−70) = −300
(iii) Subtraction tells the movement needed to go from the number being subtracted to the other number. If that starting number is negative, you must first climb up to 0 and then go on; the climb to 0 equals the size of the negative number, so it gets added. That is why (+150) − (−80) is the same as (+150) + (+80).
(iv) Brahmagupta’s rule for subtraction: subtracting a negative number is the same as adding the matching positive number. Another example: 6 − (−11) = 6 + 11 = 17.
In simple words: Going from a negative number up to a positive one means crossing 0, so the two distances add.

Teacher's Note:
1. Only 0 needs to be marked on an unmarked line.
2. Show both jumps clearly in the sketch.

Question 9. The lift of a mine moves up and down a shaft. The ground level is 0 and the levels are marked in metres above and below the ground, as in the figure.
The cage starts at level +60 m. It then moves −140, then −75, and finally +200.
(i) Find the level of the cage after each of the three moves.
(ii) Find the total movement by adding the three movements. Check it against “Target Level − Starting Level”.
(iii) Which move takes the cage below the ground for the first time?
(iv) How far above the deepest level reached is the final level of the cage?
Answer:
Rule used: Starting Level + Movement = Target Level.
(i) After the first move: (+60) + (−140) = −80 m
After the second move: (−80) + (−75) = −155 m
After the third move: (−155) + (+200) = +45 m
(ii) Total movement: (−140) + (−75) + (+200) = (−215) + 200 = −15.
Check: Target − Start = (+45) − (+60) = −15. The two agree, so the cage ends 15 m lower than where it began.
(iii) The first move (−140) does it: it takes the cage from +60 m, above the ground, to −80 m, below the ground.
(iv) The deepest level was −155 m and the final level is +45 m: (+45) − (−155) = 45 + 155 = 200.
The final level is 200 m above the deepest point.
In simple words: The cage goes to −80, −155 and then +45, so it ends 15 m lower than it started and 200 m above its lowest point.

Teacher's Note:
1. Keep a running level after each move.
2. Total movement can be checked as target minus start.

Question 10. Two players play an integer game on a board numbered from −50 to +50. Each player starts at 0. In a turn a player rolls two dice together: one die shows a number from +1 to +6 and the other shows a number from −1 to −6. The player moves by the sum of the two numbers rolled; a positive sum moves the pawn towards +50 and a negative sum towards −50.
Rohit’s first three turns are (+5, −2), then (+1, −6), then (+4, −4).
(i) Find Rohit’s move in each turn and his position after each turn.
(ii) What is the greatest forward move and the greatest backward move possible in this game?
(iii) When does a turn leave a player exactly where they were? Explain.
(iv) After the three turns, how far is Rohit from +50? Write this as a subtraction of integers.
Answer:
(i) Turn 1: (+5) + (−2) = +3; position 0 + 3 = +3.
Turn 2: (+1) + (−6) = −5; position (+3) + (−5) = −2.
Turn 3: (+4) + (−4) = 0; position (−2) + 0 = −2.
So after three turns Rohit’s pawn is at −2.
(ii) Greatest forward move: (+6) + (−1) = +5. Greatest backward move: (+1) + (−6) = −5. A pawn never moves more than 5 steps in one turn.
(iii) When the two numbers rolled are additive inverses (+1 with −1, +2 with −2, … , +6 with −6). They cancel like a zero pair, the sum is 0, and the pawn does not move.
(iv) Distance = Target − Start = (+50) − (−2) = 50 + 2 = 52. Rohit is 52 steps from +50 (but only 48 steps from −50).
In simple words: Rohit ends at −2, so he is 52 steps from +50.

Teacher's Note:
1. Each turn is a sum of a positive and a negative number.
2. Equal sizes cancel to 0.

Case Study Based Questions

Question 11. A new shopping mall in the town has a lift with buttons for every level.
Below the ground there are three parking levels, one under the other. The first one below the ground is numbered −1, the next −2 and the deepest one −3. The ground level, where the main entrance and the ticket counter are, is numbered 0. Above the ground there are five floors, numbered +1 to +5: clothes on +1, books on +2, a food court on +3, a games zone on +4 and a cinema on +5. The lift has a + button to go up and a − button to go down, and each press moves it by one level.
(i) Write the integer that stands for the second parking level below the ground.
(ii) Priya parks her scooter on level −3 and goes to the clothes floor. How many times must she press the + button?
(iii) Arun is on the food court floor and presses −5. Which level does he reach? Write an expression for this.
(iv) Which level is lower, −1 or −3? Give a reason.
Answer:
(i) −2
(ii) From −3 to +1: Movement needed = (+1) − (−3) = 1 + 3 = +4. She presses the + button 4 times.
(iii) The food court is Floor +3: (+3) + (−5) = −2. Arun reaches level −2, the second parking level.
(iv) Level −3 is lower. Both are below the ground, but −3 lies farther to the left of 0 on the number line, so −3 < −1.
In simple words: Parking levels are negative; the deeper the level, the smaller the number.

Teacher's Note:
1. Target minus start gives the number of presses.
2. Lower floors have smaller numbers.

Teacher's Note:
1. 0 °C is not below freezing.
2. Warmer minus colder gives the difference.

Question 13. Integers have a long history, and a timeline helps us see it.
Negative numbers were first used in Asia, mostly for accounting. Kautilya wrote about credit and debit in the Arthashastra around 300 BCE. In China, The Nine Chapters on Mathematical Art used red and black rods for positive and negative numbers; it was completed by about 150 CE. In India, the Bakhshali manuscript of about 300 CE wrote a negative number with a special symbol placed after the number. Then, in 628 CE, Brahmagupta gave clear rules for positive numbers, negative numbers and zero in his Brahma-sphuta-siddhanta. On a timeline we may write CE years as positive integers and BCE years as negative integers.
(i) Write the year of the Arthashastra as an integer.
(ii) Which of the four works listed is the oldest?
(iii) Arrange the four years, written as integers, in increasing order.
(iv) How many years after the Bakhshali manuscript did Brahmagupta write his book?
Answer:
(i) About 300 BCE is written as −300.
(ii) The Arthashastra is the oldest, as it is the only one from before the Common Era.
(iii) −300 < +150 < +300 < +628
(iv) 628 − 300 = 328, so Brahmagupta wrote his book 328 years after the Bakhshali manuscript.
In simple words: BCE dates are negative on a timeline, so the Arthashastra (−300) is the oldest.

Teacher's Note:
1. The smallest integer is the earliest year.
2. Both years in (iv) are CE.

Question 14. Bela bought a tall building, filled it with attractions and called it the Building of Fun.
The entry is at the ground floor, called the Welcome Hall, and it is numbered Floor 0. The building has six floors above the ground, numbered +1 to +6, and five floors below the ground, numbered −1 to −5. The Food Court is on Floor +1, the Art Centre on Floor +2, the Book Store on Floor +3 and the Sports Centre on Floor +5. Going down from the Welcome Hall you come to the Toy Store on Floor −1, the Video Games shop on Floor −2 and the Cinema Centre on Floor −3. The lift takes you up with the + button and down with the − button.
(i) Write the floor numbers of the Video Games shop and of the Book Store.
(ii) Who is on the lower floor, a child in the Toy Store or a child in the Cinema Centre?
(iii) Which button press takes you from the Art Centre to the Sports Centre?
(iv) How many floors does the building have in all, counting the ground floor?
Answer:
(i) Video Games shop: Floor −2; Book Store: Floor +3. The minus sign shows two floors below the ground and the plus sign three floors above it.
(ii) The child in the Cinema Centre (Floor −3) is lower than the child in the Toy Store (Floor −1), because −3 < −1.
(iii) Movement needed = (+5) − (+2) = +3, so press +3 (the + button three times).
(iv) 6 floors above + 5 floors below + the ground floor = 6 + 5 + 1 = 12 floors, numbered from −5 to +6.
In simple words: Floors run from −5 to +6, which is 12 floors including the ground floor.

Teacher's Note:
1. Do not forget to count Floor 0.
2. Lower floor = smaller number.

Question 15. A research submarine studies life in the deep sea.
The surface of the sea is taken as level 0. A depth of 120 m below the surface is written as −120 m, and the height of a bird flying above the sea is written with a plus sign. One morning the submarine was resting at −120 m. It then dived 85 m further down to take photographs of a coral bed. After an hour it rose 150 m to meet a diver. A sea bird was flying 25 m above the surface at the same moment.
(i) Write the height of the sea bird as an integer.
(ii) Which level did the submarine reach after the dive of 85 m?
(iii) After rising 150 m, is the submarine above or below the surface, and at which level?
(iv) How much further must the submarine rise from there to reach the surface?
Answer:
(i) 25 m above the surface is +25 m.
(ii) (−120) + (−85) = −205. The submarine reached −205 m, that is 205 m below the surface.
(iii) (−205) + (+150) = −55. The level is negative, so the submarine is still below the surface, at −55 m.
(iv) Movement needed = 0 − (−55) = +55. It must rise 55 m more.
In simple words: The submarine goes to −205 m, then up to −55 m, so it needs 55 m more to surface.

Teacher's Note:
1. Diving adds a negative; rising adds a positive.
2. The surface is level 0.

Question 16. A school held a quiz with 20 questions and used negative marking.
Every correct answer was given +4 marks. For every wrong answer 2 marks were taken away, which is written as −2. A question that was not attempted got 0. The scores could therefore be positive, zero or even negative. Ritu answered 9 questions correctly, got 6 wrong and left 5 questions unattempted. Her friend Danish answered 5 correctly, got 11 wrong and left 4 unattempted. The teacher added the marks like integers, using zero pairs to cancel the plus marks against the minus marks.
(i) Write the integer that stands for one wrong answer.
(ii) How many marks did Ritu lose for her wrong answers?
(iii) Find Ritu’s total score.
(iv) Find Danish’s total score and say what his score tells you.
Answer:
(i) −2, because 2 marks are taken away.
(ii) 6 wrong answers × 2 = 12 marks lost, written as −12.
(iii) Correct answers: 9 × 4 = +36; unattempted: 0. Score = (+36) + (−12) + 0 = +24 marks.
(iv) Correct answers: 5 × 4 = +20; wrong answers: 11 × 2 = 22 marks lost, that is −22. Score = (+20) + (−22) = −2 marks.
A negative score means he lost more marks than he won; he would have done better by leaving the doubtful questions unattempted.
In simple words: Ritu scores 24, but Danish's wrong answers cost more than he earned, giving −2.

Teacher's Note:
1. Unattempted questions add 0.
2. Find the plus and minus totals separately, then combine.

Question 17. In a copper mine, minerals are dug out both above and below the ground level.
The ground level is marked 0. A level 85 m below the ground is written as −85 m and a platform 30 m above the ground is written as +30 m. A fast lift carries workers and ore up and down the shaft. On Monday the main ore body was being cut at level −160 m, a second team was working at level −85 m, and the loading platform where the trucks wait is at level +30 m. The lift uses the same rule as any lift: Target Level − Starting Level = Movement needed.
(i) Write an integer for a level 160 m below the ground.
(ii) Which of the two working levels, −85 m or −160 m, is deeper?
(iii) A worker goes from the loading platform to the main ore body. What movement does the lift make?
(iv) From the main ore body the lift then goes to level −40 m. What movement is this?
Answer:
(i) −160 m
(ii) −160 m is deeper, because −160 < −85; it lies farther from 0 on the negative side.
(iii) Movement needed = (−160) − (+30) = −190. The lift goes down 190 m.
(iv) Movement needed = (−40) − (−160) = −40 + 160 = +120. The lift goes up 120 m.
In simple words: Down from +30 to −160 is −190 m, and up from −160 to −40 is +120 m.

Teacher's Note:
1. Always subtract the starting level from the target.
2. The sign of the answer tells up or down.

Question 18. A trekking team takes its base camp as the reference point for all heights.
Every other spot is given a height measured from the base camp, so the base camp itself is at 0 m. A spot higher than the base camp gets a positive number and a spot lower than it gets a negative number. On the map the team has marked Camp I at +450 m, Camp II at +900 m and the summit at +1600 m. Below the base camp are a lake at −120 m and a river crossing at −250 m.
(i) Write the height of the river crossing as an integer.
(ii) Which of the five marked spots is the lowest?
(iii) How much must the team climb to go from the river crossing up to Camp I?
(iv) Find the difference in height between the summit and the river crossing.
Answer:
(i) −250 m
(ii) The river crossing, since −250 is the smallest of +1600, +900, +450, −120 and −250.
(iii) Climb = (+450) − (−250) = 450 + 250 = 700 m: 250 m up to the base camp and then 450 m more.
(iv) (+1600) − (−250) = 1600 + 250 = 1850 m. The summit is 1850 m higher than the river crossing.
In simple words: The base camp is the zero here; heights below it are negative.

Teacher's Note:
1. The reference point need not be sea level.
2. Crossing 0 means adding the two distances.

Teacher's Note:
1. Leftover colour = sign of the floor.
2. Check (iv): 7 + 4 = 11 tokens.

Question 20. A food laboratory tested how fast a new freezer cools down.
The freezer was switched on when the air inside it was at +8 °C. The students watched the thermometer and found that the temperature fell by the same amount, 3 °C, in every hour. They wrote the readings as a sequence of integers, hour after hour, and were surprised to see the numbers cross zero and go on to the other side of zero. The test was stopped the moment the thermometer showed −7 °C, which is the temperature at which the food inside is safely frozen.
(i) Write the temperature inside the freezer after 2 hours.
(ii) Write the readings for the first four hours as a sequence of integers.
(iii) After how many hours did the reading first go below 0 °C?
(iv) For how many hours did the test run, and what was the total fall in temperature during it?
Answer:
(i) 8 − 3 − 3 = +2 °C.
(ii) After hours 1, 2, 3 and 4: +5, +2, −1, −4 (°C); each term is 3 less than the one before.
(iii) −1 °C is the first negative reading, and it comes after 3 hours.
(iv) After 5 hours the reading is −7 °C, so the test ran for 5 hours.
Total fall = (+8) − (−7) = 8 + 7 = 15 °C, which agrees with 5 × 3 = 15 °C.
In simple words: Falling 3 °C an hour from 8 °C, the freezer reaches −7 °C after 5 hours, a fall of 15 °C.

Teacher's Note:
1. Subtract 3 for each hour.
2. Check the total fall with 5 × 3.