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

Short Answer Questions

Question 1. In earlier classes you drew only a number ray that started at 0 and went on to the right.
(i) What kind of numbers must be added, and on which side of 0, to turn this ray into a complete number line?
(ii) Write the three integers that lie just to the left of 0.
Answer:
(i) Negative numbers must be added, placed to the left of 0. Then the ray becomes a complete number line going on for ever in both directions.
(ii) The three integers just to the left of 0 are −1, −2 and −3.
In simple words: Put the negative numbers on the left of 0 and the ray becomes a full number line.

Teacher's Note:
1. The nearest negative integer to 0 is −1.
2. A number line has arrows at both ends.

Question 2. Sunil is standing in the Book Store of Bela’s Building of Fun, which is Floor +3. He presses −6 in the lift. Write an expression of the form “Starting Floor + Movement = Target Floor” for this ride and state the floor he reaches.
Answer:
Starting Floor + Movement = Target Floor:
(+3) + (−6) = −3
Sunil goes six floors down and reaches Floor −3, the Cinema Centre, three floors below the ground.
In simple words: From Floor +3, six floors down takes Sunil to Floor −3.

Teacher's Note:
1. 3 floors down reach 0; 3 more reach −3.
2. Write the movement with its sign.

Question 3. Answer both parts about additive inverses.
(i) Write the additive inverse of +12, of −13 and of 0.
(ii) Explain in one sentence what happens when a number is added to its additive inverse.
Answer:
(i) The additive inverse of +12 is −12, of −13 is +13, and of 0 is 0 itself.
(ii) A number added to its additive inverse always gives 0, for example (+12) + (−12) = 0; in the lift, you come back to the floor you started from.
In simple words: A number and its inverse cancel each other to give 0.

Teacher's Note:
1. 0 is its own additive inverse.
2. Inverses are mirror images about 0.

Question 4. Gurmit is in the Toy Store, Floor −1. He wants to go down, but by mistake he presses the + button twice. He then presses the − button three times.
(i) Write the two button presses as integers and combine them to get the total movement.
(ii) On which floor does Gurmit finally stop?
Answer:
(i) Pressing + twice is +2 and pressing − three times is −3.
Total movement = (+2) + (−3) = −1, that is one floor down.
(ii) Starting Floor + Movement = Target Floor: (−1) + (−1) = −2.
Gurmit stops on Floor −2, the Video Games shop.
In simple words: Up 2 and down 3 is down 1, so Gurmit goes from Floor −1 to Floor −2.

Teacher's Note:
1. Combine the presses into one movement first.
2. Then add the movement to the starting floor.

Question 5. One evening the temperature in Munsiyari was 6 °C. By midnight it had come down to −3 °C. Find the fall in temperature and show your working as a subtraction of integers.
Answer:
Fall = Starting reading − Final reading
(+6) − (−3) = 6 + 3 = 9
Subtracting −3 is the same as adding +3.
So the temperature fell by 9 °C.
In simple words: 6 degrees down to 0 °C and 3 more below it is a fall of 9 °C.

Teacher's Note:
1. Count on a thermometer to check.
2. A fall is written as a positive size.

Question 6. Write an integer for each of these statements.
(i) A credit of Rs 250 in a bank passbook
(ii) A debit of Rs 180 in a bank passbook
(iii) A point 120 m below sea level
(iv) A temperature 3 °C above the freezing point of water
Answer:
Gains and 'above' are positive; losses and 'below' are negative.
(i) +250
(ii) −180
(iii) −120
(iv) +3
In simple words: Credit and above are positive; debit and below are negative.

Teacher's Note:
1. Look for the key word in each statement.
2. Keep the units in mind: Rs, m, °C.

Question 7. Arrange −7, 3, 0, −12, 5, −1 in increasing order. Then name the smallest number of the set and give a reason using the number line.
Answer:
Increasing order: −12 < −7 < −1 < 0 < 3 < 5
The smallest number is −12.
On the number line −12 lies farthest to the left, and a number on the left is always smaller than a number on its right.
In simple words: −12 is the leftmost number, so it is the smallest.

Teacher's Note:
1. Place negatives first, then 0, then positives.
2. The biggest digits on the negative side are the smallest numbers.

Question 8. Brahmagupta gave rules for adding integers in 628 CE. State his rule for adding two negative numbers and his rule for adding a number to zero, and give one example of each.
Answer:
Sum of two negatives: add the two numbers without their signs and put a minus sign in front of the answer. Example: (−4) + (−6) = −10.
A number added to zero: the answer is the same number. Examples: (−7) + 0 = −7 and 0 + (+5) = +5.
In simple words: Two negatives add to a bigger negative, and adding zero changes nothing.

Teacher's Note:
1. Give an example for each rule.
2. Zero leaves every number unchanged under addition.

Question 9. A lift in a tall building is at Floor −4 and has to reach Floor +3.
(i) Which button press is needed? Write it using the rule “Target Floor − Starting Floor = Movement needed”.
(ii) Through how many floors does the lift travel?
Answer:
(i) Movement needed = (+3) − (−4) = 3 + 4 = +7. The button press needed is +7.
(ii) The lift goes up through 7 floors: 4 floors up to Floor 0 and then 3 floors more above it.
In simple words: From −4 to +3 is 4 floors up to 0 and 3 more, so +7.

Teacher's Note:
1. Target minus start gives the press.
2. Check by counting floors.

Question 10. You have to work out (+4) − (+7) with tokens. Put down 4 positive tokens.
(i) Why can you not take away 7 positive tokens straight away, and how many zero pairs must you put down?
(ii) Write the tokens that are left and hence the value of (+4) − (+7).
Answer:
(i) Only 4 positive tokens are there, so 7 cannot be taken away. Put down 3 zero pairs; this does not change the value, and now there are 7 positive tokens and 3 negative tokens.
(ii) Take away the 7 positive tokens. 3 negative tokens are left, so (+4) − (+7) = −3.
In simple words: Add 3 zero pairs, remove 7 positives, and 3 negatives remain: −3.

Teacher's Note:
1. Zero pairs needed = tokens short.
2. Check: (+4) + (−7) = −3.

Question 11. Show how tokens are used to find (+3) − (−5). Mention how many zero pairs you must put down and what is left at the end. Also write the addition statement that gives the same answer.
Answer:
Put down 3 positive tokens. There are no negative tokens, but 5 have to be taken away.
Put down 5 zero pairs. Now there are 8 positive and 5 negative tokens, and the value is still +3.
Take away the 5 negative tokens. 8 positive tokens are left, so (+3) − (−5) = +8.
Addition statement: (+3) + (+5) = +8, because subtracting a negative number is the same as adding the matching positive number.
In simple words: Add 5 zero pairs, take away the 5 negatives, and 8 positives remain.

Teacher's Note:
1. Removing negatives leaves extra positives.
2. Both methods must give the same answer.

Question 12. Four shopkeepers close the day with these balances: Asha Rs −250, Babu Rs +180, Chandra Rs −40 and Dev Rs 0.
(i) Arrange the four balances in increasing order.
(ii) Whose account is the worst off, and whose is neither in credit nor in debit? Give a reason for each.
Answer:
A negative balance means money is owed and a positive balance means money in hand.
(i) −250 < −40 < 0 < +180, that is Asha, Chandra, Dev, Babu.
(ii) Asha is the worst off, because −250 is the smallest balance, so she owes the most. Dev’s account is neither in credit nor in debit, because his balance is 0, which is neither positive nor negative.
In simple words: Asha owes the most; Dev has exactly zero.

Teacher's Note:
1. Order on the number line from left to right.
2. Zero balance means nothing owed and nothing saved.

Question 13. In a geographical cross section, point P is at a height of +80 m and point Q is at a height of −250 m.
(i) Which point is higher, and what is the reference level used here?
(ii) How much higher is it? Show the subtraction.
Answer:
(i) Point P is higher, because +80 is above 0 while −250 is below 0. The reference level is sea level, taken as 0 m.
(ii) (+80) − (−250) = 80 + 250 = 330. So P is 330 m higher than Q.
In simple words: P is 80 m above sea level and Q is 250 m below it, so P is 330 m higher.

Teacher's Note:
1. Higher minus lower gives the difference.
2. Subtracting a negative adds.

Question 14. A beetle sits at −9 on a number line and makes equal jumps of +5 each.
(i) Write the number it lands on after the first, second and third jump.
(ii) After how many jumps is the beetle standing on a positive number for the first time?
Answer:
Each jump adds 5 to the position.
(i) First jump: (−9) + 5 = −4; second jump: (−4) + 5 = +1; third jump: (+1) + 5 = +6.
(ii) The first positive number reached is +1, so the beetle is on a positive number after 2 jumps.
In simple words: −9, −4, +1, +6: the beetle first reaches a positive number after 2 jumps.

Teacher's Note:
1. Add 5 each time.
2. Positive means greater than 0.

Question 15. Look at the sequence −27, −21, −15, −9, …
(i) Write the rule of the pattern.
(ii) Write the next four terms and state which of them is the first positive term.
Answer:
(i) The rule is “add 6 to get the next term”, since (−27) + 6 = −21 and (−21) + 6 = −15.
(ii) The next four terms are −3, +3, +9, +15. The first positive term among them is +3.
In simple words: Keep adding 6: −3, 3, 9, 15; the first positive term is 3.

Teacher's Note:
1. Check the gap between every pair of terms.
2. 0 is skipped in this pattern.

Question 16. Priya says that there are exactly 6 integers lying between −12 and −6.
(i) Write all the integers between −12 and −6 in increasing order.
(ii) Is Priya right? Give the correct count and explain where she went wrong.
Answer:
The integers between −12 and −6 do not include −12 and −6 themselves.
(i) −11, −10, −9, −8, −7
(ii) Priya is not right. There are only 5 such integers. She has counted one of the end numbers, −12 or −6, along with the integers between them.
In simple words: Only 5 integers lie between −12 and −6; Priya counted an end number.

Teacher's Note:
1. 'Between' leaves out both ends.
2. Listing the numbers avoids counting mistakes.

Question 17. For each of the following, say whether the answer is always positive, always negative, or can be of either sign. Give one example to support each answer.
(i) (negative) + (negative)
(ii) (positive) − (negative)
(iii) (positive) + (negative)
(iv) (positive) − (positive)
Answer:
Think of each as a movement in the lift.
(i) Always negative: both are moves downwards, e.g. (−3) + (−5) = −8.
(ii) Always positive: subtracting a negative is adding a positive, e.g. (+2) − (−6) = +8.
(iii) Either sign: (+7) + (−2) = +5 but (+2) + (−7) = −5; it is 0 when the numbers are inverses, as in (+3) + (−3) = 0.
(iv) Either sign: (+9) − (+5) = +4 but (+5) − (+9) = −4; it is 0 when the two numbers are equal.
In simple words: Down plus down is down, positive minus negative is up, and the other two depend on which number is larger.

Teacher's Note:
1. One counter-example is enough for 'either sign'.
2. Use the lift picture to reason.

Teacher's Note:
1. Find the border sum from a complete line first.
2. Always check with another line.

Question 19. Four events from the history of integers are: Kautilya’s Arthashastra about 300 BCE, The Nine Chapters on Mathematical Art about 150 CE, the Bakhshali manuscript about 300 CE and Brahmagupta’s Brahma-sphuta-siddhanta in 628 CE.
(i) Write each year as an integer, taking CE years as positive and BCE years as negative.
(ii) Arrange the four integers in increasing order.
(iii) How many years after the Bakhshali manuscript was Brahmagupta’s book written?
Answer:
(i) Arthashastra: −300; The Nine Chapters: +150; Bakhshali manuscript: +300; Brahma-sphuta-siddhanta: +628.
(ii) −300 < +150 < +300 < +628
(iii) 628 − 300 = 328, so Brahmagupta’s book was written 328 years after the Bakhshali manuscript.
In simple words: On a timeline BCE is negative and CE is positive; 628 − 300 = 328 years.

Teacher's Note:
1. BCE years go to the left of 0 on a timeline.
2. Both dates in (iii) are CE, so just subtract.

Teacher's Note:
1. Strike out the row and column each time, or you may pick two numbers from one line.
2. The magic is in how the grid is built.