CBSE Class 7 Mathematics Number Play Worksheet Set 01

Very Short Answer Questions

Question 1. Children stand one behind the other and each calls out the number of children in front of them who are taller. One child calls out 3. What does this tell us about the children standing in front of that child?
Answer:
There are at least three children in front of that child, and exactly three of them are taller than the child.

Teacher's Note:
1. There may be more children ahead who are shorter.
2. Only the taller ones are counted.

Question 2. In the same game, a child who is not at the front of the line calls out 0. What does this tell you about that child?
Answer:
That child is taller than every child standing in front, though some child behind may still be taller.

Teacher's Note:
1. 0 means nobody ahead is taller.
2. It does not mean the child is the tallest of all.

Question 3. Can a collection of an odd number of dots be arranged completely in pairs? Give a reason.
Answer:
No. An odd number always leaves one dot over when the dots are put in pairs; only an even number pairs up fully.

Teacher's Note:
1. This is the meaning of 'odd'.
2. Try 7 dots: 3 pairs and 1 left over.

Question 4. Write the parity of the sum 12 + 18 + 7.
Answer:
12 + 18 + 7 = 37. Since even + even + odd = odd, the sum is odd.

Teacher's Note:
1. One odd number in a sum makes it odd.
2. Check: 37 ends in 7.

Question 5. Is the sum of 6 odd numbers even or odd? Give the reason.
Answer:
Even. Six odd numbers leave six extra ones, and these make three more pairs, so the whole sum pairs up fully.

Teacher's Note:
1. An even count of odd numbers gives an even sum.
2. Example: 1 + 1 + 1 + 1 + 1 + 1 = 6.

Question 6. Using the rule for odd numbers, find the 75th odd number.
Answer:
The nth odd number is 2n − 1, so the 75th odd number = 2 × 75 − 1 = 149.

Teacher's Note:
1. The 75th even number is 150.
2. The odd number is one less.

Question 7. A 3 × 3 grid is to be filled with the numbers 1 to 9 without repeating any number. As stated in the chapter, in how many different ways can this be done?
Answer:
It can be done in 3,62,880 different ways.

Teacher's Note:
1. This is 9 × 8 × 7 × … × 1.
2. Yet there is essentially only one magic square.

Question 8. For such a grid, the sum of each row and of each column is written in a circle outside the grid. What is the total of all six circled numbers?
Answer:
The three row sums add to 45 and the three column sums also add to 45, so the total is 90.

Teacher's Note:
1. Each set of three sums uses every number once.
2. 45 + 45 = 90.

Question 9. Name the Italian mathematician who wrote about the sequence 1, 2, 3, 5, 8, 13, … in 1202 CE, about 500 years after Virahāṅka.
Answer:
Fibonacci, who wrote about these numbers in 1202 CE.

Teacher's Note:
1. That is why they are called Virahāṅka–Fibonacci numbers.
2. Virahāṅka came first.

Question 10. In the poetry of Indian languages, for how many beats does a long syllable last as compared with a short syllable?
Answer:
A long syllable lasts 2 beats, twice as long as a short syllable, which lasts 1 beat.

Teacher's Note:
1. Short = 1 beat, long = 2 beats.
2. Rhythms are sums of 1s and 2s.

Question 11. Give one example from nature in which Virahāṅka numbers are seen.
Answer:
The number of petals on a daisy is usually a Virahāṅka number, such as 13, 21 or 34.

Teacher's Note:
1. Sunflower seed spirals are another example.
2. Give actual numbers in the answer.

Question 12. In the cryptarithm A4 + A4 = BCC, each letter stands for one digit. Find A, B and C.
Answer:
The sum is at most 94 + 94 = 188 < 200, so B = 1; also 4 + 4 = 8, so C = 8, and the tens must give 18, so A = 9.
Only 94 + 94 = 188 fits, so A = 9, B = 1, C = 8.

Teacher's Note:
1. 4 + 4 = 8 fixes C = 8 at once.
2. Then 2 × A must end in 8 too.

Question 13. Five children stand in a line and the numbers called out, from the front, are 0, 1, 2, 0, 3. Is such an arrangement possible? Give a reason.
Answer:
Yes. The child at place n has n − 1 children ahead, so the limits are 0, 1, 2, 3, 4.
Here 0 ≤ 0, 1 ≤ 1, 2 ≤ 2, 0 ≤ 3 and 3 ≤ 4, so every number is allowed.

Teacher's Note:
1. Compare each number with the children ahead.
2. None of the numbers is too big.

Question 14. Four children stand in a line. Explain why they can never call out 0, 1, 1, 4.
Answer:
The fourth child has only three children in front, so that child can call out at most 3, never 4.

Teacher's Note:
1. A child cannot count more children than are ahead.
2. Position 4 allows at most 3.

Question 15. Can four odd numbers add up to 35? Give a reason using parity.
Answer:
No. An even number of odd numbers always adds to an even number, but 35 is odd.

Teacher's Note:
1. Four odd numbers make two extra pairs.
2. So the sum is even.

Question 16. Two cousins were born exactly one year apart, and today is the birthday of both. Can the sum of their ages be 48? Give a reason.
Answer:
No. Their ages are consecutive numbers, one even and one odd, so the sum is always odd; 48 is even.

Teacher's Note:
1. Consecutive numbers have different parities.
2. Even + odd = odd.

Question 17. Without multiplying, state the parity of the number of small squares in a 245 × 316 grid, and give the rule you used.
Answer:
Even. A product is odd only when both factors are odd; here 316 is even, so the product is even.

Teacher's Note:
1. One even factor is enough.
2. No need to multiply.

Question 18. Does the expression 5n + 2 always give an even value, always an odd value, or either? Give one value of n to support your answer.
Answer:
Either. For n = 1, 5n + 2 = 7 (odd); for n = 2, 5n + 2 = 12 (even).

Teacher's Note:
1. The parity of 5n follows the parity of n.
2. Two values show both cases.

Question 19. A 3 × 3 grid is filled with the numbers 1 to 9 without repetition. Explain why a circled row sum of 26 is impossible.
Answer:
The largest sum of three different numbers from 1 to 9 is 7 + 8 + 9 = 24, so a row sum of 26 can never occur.

Teacher's Note:
1. Find the largest possible row sum first.
2. 26 is more than 24.

Question 20. A 3 × 3 magic square is made from nine consecutive numbers and the number at its centre is 25. What is its magic sum?
Answer:
Every line adds to three times the centre number, so the magic sum = 3 × 25 = 75.

Teacher's Note:
1. Magic sum = 3 × centre.
2. Check with 1 to 9: 3 × 5 = 15.

Question 21. Two numbers standing next to each other in the Virahāṅka sequence are 144 and 233. Write the next two numbers of the sequence.
Answer:
144 + 233 = 377 and 233 + 377 = 610, so the next two numbers are 377 and 610.

Teacher's Note:
1. Always add the last two numbers.
2. Use the new number for the next step.

Question 22. In how many different ways can the number 4 be written as a sum of 1s and 2s?
Answer:
In 5 ways: 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 2 + 1, 2 + 1 + 1 and 2 + 2.

Teacher's Note:
1. Order matters: 1 + 2 + 1 and 2 + 1 + 1 are different.
2. 5 is the 4th Virahāṅka number.

Question 23. Solve the cryptarithm JK + JK = KLL, where JK is a two-digit number and KLL is a three-digit number.
Answer:
The answer is less than 200, so K = 1; then 2 × J1 = 1LL gives J = 6, L = 2.
J = 6, K = 1, L = 2, because 61 + 61 = 122.

Teacher's Note:
1. The hundreds digit of the sum fixes K.
2. Then 1 + 1 = 2 gives L.

Question 24. Solve the cryptarithm M1 + M = 1NN, where M1 is a two-digit number, M is a one-digit number and 1NN is a three-digit number.
Answer:
To reach at least 100, M1 + M must be large, so M = 9.
M = 9 and N = 0, because 91 + 9 = 100.

Teacher's Note:
1. The largest possible sum is 91 + 9 = 100.
2. So the answer is exactly 100.

Question 25. A tap is closed. Its handle is turned 46 times, and each turn changes it from closed to open or from open to closed. Is the tap finally open or closed? Give the parity reason.
Answer:
Closed. Every two turns bring the tap back to where it was; 46 is even, so it ends in its starting state.

Teacher's Note:
1. Even number of changes: same state.
2. Odd number of changes: opposite state.

Question 26. In a line of six children the numbers called out, from the front, are 0, 1, 1, 1, 1, 1. At which place in the line does the tallest child stand? Give a reason.
Answer:
The tallest child always calls out 0, and only the first child calls out 0.
So the tallest child stands first, at the front.

Teacher's Note:
1. Look for the 0s in the list.
2. The other five each have exactly one taller child ahead: the first child.

Question 27. Every number of a 3 × 3 magic square made with 1 to 9 is increased by 10. Is the result still a magic square, and what is its magic sum?
Answer:
Yes, it is still a magic square. Each line of three numbers gains 3 × 10 = 30.
New magic sum = 15 + 30 = 45.

Teacher's Note:
1. Every line gains the same amount.
2. Add 10 three times, not once.

Question 28. What is the parity of the 30th number of the Virahāṅka sequence? Give the reason.
Answer:
The parities repeat as odd, even, odd, so the even numbers are at places 2, 5, 8, …, which leave remainder 2 when divided by 3.
30 leaves remainder 0, so the 30th number is odd.

Teacher's Note:
1. Find the remainder when the place is divided by 3.
2. Remainder 2 means even; otherwise odd.

Question 29. Solve the cryptarithm DE + ED = FEF, where DE and ED are two-digit numbers and FEF is a three-digit number.
Answer:
The sum is below 200, so F = 1. Then the units give D + E ending in 1, and the tens give E = 2, D = 9.
D = 9, E = 2, F = 1, because 92 + 29 = 121.

Teacher's Note:
1. Start with the hundreds digit.
2. Always check by adding.

Question 30. The numbers 1 to 100 are written down and one of them is rubbed out. The remaining 99 numbers add up to an odd number. Was the rubbed-out number odd or even? Give the reason.
Answer:
1 + 2 + … + 100 = 5050, which is even.
Even − odd = odd, so the rubbed-out number was odd.

Teacher's Note:
1. Taking away an even number keeps the parity.
2. Taking away an odd number changes it.