How to Spot Number Patterns and Boost Your Maths Skills
Mathematics is a subject that helps to increase one’s reasoning ability and challenges their problem-solving skills to a certain extent. It is a medium that helps to route our critical thoughts accurately.
However, many of us find specific chapters tricky and confusing, such as permutation and combination, geometry and playing with numbers, etc. It happens because we lack in-depth knowledge about these concepts.
However, if you are dealing with problematic mathematical topics seek playing with numbers Class 6 NCERT solutions! We hope the solved answers will be beneficial for the exam guidance.
Let’s start gathering knowledge!
Solved Problem on Playing with Numbers Class 6 - Exercise 3.7
If you are studying class 8 playing with numbers, we would advise you to go through the following solutions. Ace your studies with these exercises!
1. Mohanty purchases two bags of mangoes, weighing 45 kg and 25 kg. Point out the maximum amount of weight for measuring the number of mangoes (number of times).
Ans: Since the weight of mangoes in two bags are present = 45 kg and 25 kg.
The maximum amount of weight will be = H.C.F. of the two bags (45, 25)
45 = 3 * 3 * 5
25 = 5 * 5
H.C.F. = 5
Therefore, 5 kg is the maximum amount of weight for two bags of mangoes (the exact number of times).
2. Two boys start stepping off from an identical place. The steps amount to 63 cm, 70 cm and 77 respectively. What will be the minimum distance that each of them should cover so that all of them can complete the entire distance? (Problem: Playing with numbers)
Ans: 1st boy’s step is = 63 cm,
2nd boy’s step is = 70 cm,
3rd boy’s step measures = 77 cm.
L.C.M. of 63, 70 and 77 using the division method
(Image to be added soon)
∴ L.C.M. = 2*3*3*5*7*11 = 6,930 cm
Therefore, the minimum required distance will be 6,930 cm to cover the entire distance.
3. Suppose there are 3 tankers, containing 50 litres, 45 litres and 60 litres of oil, respectively. Find out the highest capacity that measures the oil of all three tankers (number of times).
Ans: Maximum amount of capacity measures through = H.C.F. of (50, 45, 60)
50 = 2*5*5
45 = 3*3*5
60 = 3*2*2*5
∴ H.C.F. = 5 (because it is the common factor)
So, the maximum capacity of the tanker will be 5 litres.
Tips to Tackle Mathematics Problems Regarding Playing with Number Class 8
You may find yourself under a lot of stress when you face complex math problems. However, you can reduce these problems through the following tips:
Try to finish your chapters along with the class. Leaving things for the last minute can worsen your condition. Solving problems related to numbers daily can help you gain confidence. You can also learn from NCERT Solutions for the class 8 maths chapter playing with numbers to score better!
Also, you should try to memorise all the vital formulas before you finish your studies. Keeping the formulas in mind will help you solve the problems quickly. Try recalling the techniques every day apart from learning Class 6 maths playing with numbers before you call it a day to score well in maths.
Students often assume mathematics as a difficult subject. However, you can score better grades in maths if you understand all the concepts clearly.
If you are facing any problem with difficult topics, try going through NCERT Solutions for Class 8 Maths Chapter 16 Exercise 16.2. You can also visit and learn from our live classes and get detailed solutions for playing with numbers Class 6 CBSE.
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In Mathematics Class 8, playing with numbers is one of the most important concepts. As the name suggests, the topic is all about numbers and their properties. There are 4 different types of numbers in mathematics. These are natural numbers, whole numbers, integers, and rational numbers.
Natural numbers are the numbers that include all positive integers excluding 0. Whole numbers are similar to the natural numbers but they also include 0. Integers include negative numbers, positive numbers and 0 also. And rational numbers are nothing but fractions in the form of a/b where b should not be equal to zero.
Read the full article to know more about playing with numbers. Here we will discuss the definition of play with numbers, the General form of numbers, Games with numbers, tests of divisibility, and solved examples in the sections below.
As it is suggested by the name the chapter is all about numbers. You can play various games with numbers. The names of the game which we are going to play in this article are reversing the digits of a number, the general form of the number, letters for digits, and tests of divisibility. The process in detail is discussed here.
Number in General Form
Any number can be represented in simple form following the simple procedure. The only thing to identify a number is to identify the position of each digit in the given number. Now multiply the position of the digit, i.e tens digit 10 * number and add the remaining digits. If we take it as an example AB, take a two-digit number AB. Its general form will be 10 * A + B. The result after applying simple multiplication and addition will be the original number.
The first step is to reverse the digits of the number. Now add the original number, and reversed number. If you divide the number which you got after sum by 11, the remainder will always be zero. You can play this game with either a two-digit number or a three-digit number. But remember here that for three-digit numbers, the procedure will be different.
Letters for Digits
Here we are going to deal with two or more numbers and have operators among them. Just perform these operations and by trial and error method find out which number suits an expression that satisfies the equation. It is one of the most interesting games to play with numbers.
Tests of Divisibility
The test of divisibility is if the given list of some numbers is divided by the particular number or not.
Number Puzzles
Playing with different types of numbers will increase your problem-solving skills as well as your reasoning skills. Here you just have to fill the given numbers in the grid or magic triangle or any other by satisfying the given conditions.
Solved Examples
1. Write the following list of numbers in the generalized form.
a. 145
b. 79
c. 67
d. 777
Ans: Given numbers are 145, 79, 59, 895
The general form of 145 is as follows:
145 can be expressed as 100 + 40 + 5
1 × 100 + 4 × 10 + 5 is 145.
79 = 70 + 9
= 7 × 10 + 9
67 = 60 + 7
= 6 × 10 + 7
777 = 700 + 70 + 7
= 7 × 100 + 7 × 10 + 7
2. Prove that the following numbers satisfy the method of reversing the digits.
a. 35
b. 99
Ans:
Here Given number is 35. Hence the reverse of 35 would be 53.
Now if we will add these numbers then
35 + 53 = 88
Here 88 is the number that we get by adding the number and the number which we get by reversing its digits. Here we can see that 88 is exactly divisible by 11.
88= 11 × 8
Hence the method for reversing the digits is proved here.
Here Given number is 99.
Hence the reverse of 99 would be 99.
Now if we will add these numbers then
99 + 99 = 198
Here 198 is the number which we get by adding the number and the number which we get by reversing its digits. Here we can see that 88 is exactly divisible by 11.
198= 11× 18
Hence the method for reversing the digits is proved here.
FAQs on Playing With Numbers: Tricks, Patterns & Solutions
1. What does it mean to write a number in its 'generalised form' in the chapter Playing with Numbers?
Writing a number in its generalised form means expressing it as a sum of its digits multiplied by their respective place values. This algebraic representation helps in solving number puzzles and proving divisibility rules. For example:
- A 2-digit number 'ab' can be written as 10a + b. For instance, 68 is 10(6) + 8.
- A 3-digit number 'abc' can be written as 100a + 10b + c. For instance, 492 is 100(4) + 10(9) + 2.
2. How do you use divisibility rules to quickly check if a large number is divisible by 3 or 9?
In Playing with Numbers, divisibility rules are shortcuts to check for division without performing the actual calculation. Here’s how to test for 3 and 9:
- Test for Divisibility by 3: A number is divisible by 3 if the sum of its digits is a multiple of 3. For example, for the number 1245, the sum is 1+2+4+5 = 12. Since 12 is divisible by 3, the number 1245 is also divisible by 3.
- Test for Divisibility by 9: A number is divisible by 9 if the sum of its digits is a multiple of 9. For the number 729, the sum is 7+2+9 = 18. Since 18 is divisible by 9, the number 729 is also divisible by 9.
3. Can you explain the 'letters for digits' puzzles from the Playing with Numbers chapter with an example?
Puzzles involving 'letters for digits', also known as cryptarithmetic, are problems where letters stand for digits in an arithmetic equation. The goal is to find which digit each letter represents. Each letter must represent a unique digit. For example, consider the addition problem:
A B
+ 3 7
------
6 A
Here, we find the value of A and B. From the ones column, B + 7 gives a number ending in A. From the tens column, A + 3 (plus any carry-over) equals 6. If we test A=2, then B+7 must end in 2, which means B=5. Let's check: 25 + 37 = 62. This fits the pattern 6A. So, A = 2 and B = 5.
4. Why does reversing a 2-digit number and adding it to the original always result in a number divisible by 11?
This number game works because of the principles of place value and algebra. Let the original 2-digit number be 'ab'.
- In its generalised form, this number is 10a + b.
- When you reverse the digits, the new number is 'ba', which in generalised form is 10b + a.
- Adding the two numbers gives: (10a + b) + (10b + a) = 11a + 11b.
- This can be factorised as 11(a + b).
Since the sum is always a product of 11 and another number (a+b), it will always be perfectly divisible by 11.
5. A number is divisible by 3. Does that mean it must also be divisible by 9? Explain why or why not.
No, a number divisible by 3 is not necessarily divisible by 9. While all numbers divisible by 9 are also divisible by 3, the reverse is not true. This is because 9 is a multiple of 3, but 3 is not a multiple of 9.
For example, the number 24 is divisible by 3 (2+4=6, which is a multiple of 3), but it is not divisible by 9 (2+4=6, which is not a multiple of 9). The rule for divisibility by 9 is stricter than the rule for 3.
6. What is the difference between a factor and a multiple, and how are these concepts used in Playing with Numbers?
Factors and multiples are fundamental concepts for understanding number properties.
- A factor is a number that divides another number completely without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
- A multiple is the result of multiplying a number by an integer. For example, multiples of 12 are 12, 24, 36, and so on.
These concepts are the basis for finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM), which help solve problems about grouping items or finding when events will happen at the same time.
7. How is the concept of place value essential to the games and puzzles in the 'Playing with Numbers' chapter?
Place value is the foundational concept upon which the entire chapter is built. It dictates that the value of a digit depends on its position (ones, tens, hundreds, etc.). This is critical for several reasons:
- Generalised Form: We can only express a number like 54 as 10x5 + 4 because we understand the place values of '5' and '4'.
- Number Games: The trick of reversing digits and adding them works only because we can algebraically manipulate their place values (10a+b becomes 10b+a).
- Divisibility Rules: The rules for divisibility by numbers like 3, 9, or 11 are derived by analysing the sum or pattern of digits based on their place value contributions.
Without understanding place value, these number tricks and rules would seem like magic instead of logical mathematical properties.
8. What are the key topics covered in the CBSE Class 8 Maths chapter 'Playing with Numbers'?
According to the CBSE syllabus for the 2025-26 session, the main topics in the Class 8 Maths chapter 'Playing with Numbers' include:
- Numbers in General Form: Representing 2-digit and 3-digit numbers algebraically (e.g., ab = 10a + b).
- Games with Numbers: Exploring patterns and properties, such as reversing digits of a number and performing arithmetic operations on them.
- Letters for Digits: Solving puzzles where letters stand for digits in arithmetic sums (cryptarithmetic).
- Tests of Divisibility: Understanding and applying the rules of divisibility for 2, 3, 4, 5, 6, 8, 9, 10, and 11.
