Guide to Common Word Problem Scenarios in Number System

Word problems in the number system often seem tricky, but with the right approach, they can become much simpler. This guide breaks down the most frequent word problem scenarios in the number system, providing actionable solutions and examples to make learning easier. Let’s simplify the number system together!

ScenarioExplanationCommon ExampleUnique Example
1. Finding the Largest/Smallest Number
Identifying the largest or smallest number in a set of numbers, intervals, or conditions.		Example: Find the smallest 4-digit number divisible by 9.
Solution: Smallest 4-digit number = 1000. Dividing 1000 by 9 gives a remainder of 1. Adding 9-1 = 8 to 1000 gives 1008 (divisible by 9). Answer: 1008.		Example: What is the largest 3-digit number where the sum of the digits equals 15?
Solution: Start with the largest number, 999, and check sums. The largest number is 960, where 9+6+0=15. Answer: 960.
2. Divisibility and RemaindersDetermining divisibility, remainders, or conditions of divisibility for specific numbers.Example: Is 144 divisible by both 6 and 9?
Solution: Divisibility rule: 144 ÷ 6 = 24 (no remainder) and 144 ÷ 9 = 16 (no remainder). Answer: Yes.		Example: A number leaves a remainder of 3 when divided by 5. What is the smallest such number greater than 30?
Solution: Start with 30+1=31. Since 31 ÷ 5 leaves the remainder 1, the next number = 33. Answer: 33.
3. Prime and Composite NumbersQuestions about identifying or using prime and composite numbers.Example: Which of the following are prime: 29, 35, 49?
Solution: 29 is divisible only by 1 and itself (prime). 35 (divisible by 5), 49 (divisible by 7) are composite. Answer: 29.		Example: Find three consecutive composite numbers greater than 90.
Solution: 90 (composite), 91 (divisible by 7), 92 (divisible by 2). Answer: 90, 91, 92.
4. LCM and HCFProblems involving finding the least common multiple or highest common factor of two or more numbers.Example: Find the LCM and HCF of 12 and 18.
Solution: LCM = 36, HCF = 6 (from prime factorization: 12=2²×3, 18=2×3²). Answer: LCM=36, HCF=6.		Example: Two gears have 15 and 25 teeth, respectively. How many revolutions will they complete before aligning?
Solution: LCM of 15 and 25 = 75. They align after 75 revolutions.
Answer: 75 revolutions.
5. Sum or Product of DigitsQuestions involving the sum, product, or properties of digits of a number.Example: Find the sum of the digits of 245.
Solution: 2 + 4 + 5 = 11.		Example: A 2-digit number has a sum of digits equal to 12, and the number exceeds 50. What is the number?
Solution: Possible pairs (6,6), (7,5), etc. Only 75 satisfies. Answer: 75.
6. Odd and Even NumbersDetermining if numbers are odd, even, or using their properties in calculations.Example: Find the sum of the first 10 even numbers.
Solution: Sum = 2+4+6+…+20
= n/2 × (first term + last term) = 10/2 × (2+20) = 110 Answer: 110.		Example: Find three consecutive odd numbers such that their sum is 99.
Solution: Let the numbers be 𝑥, 𝑥+2, 𝑥+4.
𝑥+𝑥+2+𝑥+4=99 →
3𝑥+6=99 →
𝑥 = 31
Answer: 31, 33, 35.
7. Square and Cube NumbersQuestions requiring identification or use of square or cube numbers.Example: Is 64 a perfect square?
Solution: $\sqrt{64}$ =8 (an integer).
Answer: Yes.		Example: A cube-shaped box has a volume of 216 cubic cm. What is the length of one side?
Solution: $\sqrt[3]{216}$ =6
Answer: 6 cm.
8. Ratio and ProportionWord problems involving ratios, proportions, or comparisons.Example: If the ratio of boys to girls in a class is 3:4 and there are 28 students, how many are boys?
Solution: Total parts = 3+4=7
Boys = 3/7 $\times$ 28 =12.
Answer: 12 boys.		Example: The ratio of red to blue marbles is 5:8. If there are 39 marbles in total, how many are blue?
Solution: Total parts =
5+8=13
Blue marbles = 8/13 $\times$ 39 = 24.
Answer: 24.
9. Recurring DecimalsConverting recurring decimals to fractions or working with their properties.Example: Convert 0.666… into a fraction.
Solution: Let 𝑥=0.666…
Multiply by 10 → 10𝑥 = 6.666… Subtract 𝑥=0.666…
: 9𝑥=6 → 𝑥 =6/9→ 𝑥 =2/3.
Answer: 2/3.		Example: Express 0.272727… as a fraction.
Solution: Let 𝑥 = 0.272727…
Multiply by 100 → 100𝑥 = 27.272727…
Subtract 𝑥 =0.272727…
99𝑥 =27 → 𝑥 = 27/99 = 3/11
Answer: 3/11.
10. Remainder TheoremDetermining properties of numbers when divided by another number.Example: What is the remainder when 56 is divided by 7?
Solution: 56 ÷ 7 = 8 with remainder 0.
Answer: 0.		Example: A number gives a remainder of 2 when divided by 9. Find the smallest such number greater than 50.
Solution: Start with 50+1=51. Check 51÷9: remainder is not 2. Next, 53. Answer: 53.
11. Sum of SeriesFinding the sum of a series of numbers (arithmetic, geometric, or special).Example: Find the sum of the first 10 natural numbers.
Solution: S=n/2 $\times$ (first term + last term) = 10/2 $\times$ (1+10) = 55
Answer: 55.		Example: Find the sum of the squares of the first 5 odd numbers.
Solution: Squares: 1², 3², 5², 7², 9²
= 1+9+25+49+81 = 165. Answer: 165.
12. Forming NumbersForming the largest or smallest number using given digits or conditions.Example: Form the largest number using the digits 3, 7, 1, and 9.
Solution: Arrange in descending order: 9731. Answer: 9731.		Example: Form the smallest 4-digit number where the sum of the digits is 15.
Solution: Distribute digits: 1+4+4+6 = 15. Arrange in ascending order: 1446. Answer: 1446.

Mastering number system word problems doesn’t have to be intimidating. With this guide, you’ll develop a structured approach to solving even the most challenging problems. Practice regularly, stay curious, and don’t hesitate to revisit this guide whenever you need clarity.

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