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NCERT Class 9 Mathematics Exercise Set 3.1 Solutions: The World of Numbers

Step-by-step working for every question of Exercise Set 3.1 in NCERT's Class 9 Mathematics book, chapter “The World of Numbers”, written by Shishya. Each question is in our own short words; open the NCERT chapter for the exact wording and figures.

1. Ex 3.1 Q1

   A Lothal merchant gets 15 copper ingots for every 2 bags of spices. How many ingots does he get for 12 bags?

   1. The ratio of copper ingots to bags of spices is 15 to 2.
   2. Set up the proportion as 15/2 = x/12, where x is the number of ingots for 12 bags.
   3. Cross-multiply to get 2 × x = 15 × 12.
      why?

      If a/b = c/d (b and d not zero), multiplying both sides by bd gives ad = bc — the same 'do the same to both sides' rule.

   4. Calculate the right side: 15 × 12 = 180.
   5. Divide both sides by 2 to find x: x = 180 ÷ 2.
      why?

      Doing the same thing to both sides of an equation (adding, subtracting, multiplying or dividing by the same non-zero number) keeps the two sides equal, so the equation stays true while the unknown is left on its own.

   6. Solve for x to get x = 90.

   Answer: 90 ingots

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2. Ex 3.1 Q2

   One column of the Ishango bone shows 11, 13, 17, 19. What do these numbers share? Give the next three numbers of the same kind.

   1. Look at the numbers 11, 13, 17, and 19.
   2. Check if 11 has any factors other than 1 and itself; it does not.
   3. Check if 13 has any factors other than 1 and itself; it does not.
   4. Check if 17 has any factors other than 1 and itself; it does not.
   5. Check if 19 has any factors other than 1 and itself; it does not.
   6. Since all these numbers have only two factors, 1 and the number itself, they are prime numbers.
   7. After 19: 20 = 2 × 10, 21 = 3 × 7 and 22 = 2 × 11 are not prime, but 23 has no factor other than 1 and 23, so 23 is prime.
   8. After 23: 24, 26 and 28 are even, 25 = 5 × 5 and 27 = 3 × 9, while 29 has no factor other than 1 and 29, so 29 is prime.
   9. After 29: 30 = 2 × 15 is not prime, and 31 has no factor other than 1 and 31, so 31 is prime.

   Answer: The numbers share the property of being prime; the next three numbers are 23, 29, 31.

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3. Ex 3.1 Q3

   Natural numbers are closed under addition. Are they closed under subtraction? Justify with examples.

   1. Natural numbers are the counting numbers 1, 2, 3, 4, …
   2. Closed under subtraction would mean: the difference of any two natural numbers is again a natural number.
   3. Sometimes it works: 8 − 3 = 5, which is a natural number.
   4. But 3 − 5 = −2, which is not a natural number.
   5. Also 7 − 7 = 0, and 0 is not a natural number either.
   6. One failing example is enough to show the property does not hold for all pairs.

   Answer: No. For example 3 − 5 = −2 and 7 − 7 = 0 are not natural numbers, so natural numbers are not closed under subtraction.

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4. Ex 3.1 Q4

   Counting on finger joints: each finger has 3 joints and the thumb points to them. How many can be counted on one hand, and how does this link to base-12 counting?

   1. Each of the four fingers (all except the thumb) has 3 joints (segments).
   2. The thumb touches the joints one by one, so one hand counts 4 × 3 = 12.
   3. So one hand naturally counts one dozen, which is why counting in twelves (base 12, as in dozens) fits this method.
   4. If the five fingers of the other hand keep track of how many dozens are done, two hands count up to 5 × 12 = 60, the base of the old sexagesimal system still used for minutes and seconds.

   Answer: 12 on one hand (4 fingers × 3 joints), which is one dozen, the unit of base-12 counting.

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