NUMERICAL ANALOGIES: TYPES, APPLICATIONS AND EXERCISES - MATHS - 2024

The number analogies refer to similarities found in the properties, meaning numerical order and arrangements where call analogy to such similarity. In most cases, a structure of premises and unknown is preserved, where a relationship or operation is verified in each one of them.

Numerical analogies usually require cognitive analysis, which obeys different types of reasoning that we will classify in depth later.

Meaning of analogy and its main types

It is understood by analogy to the similar aspects presented between different elements, these similarities can be presented in any characteristic: Type, shape, size, order, context, among others. We can define the following types of analogy:

• Numerical analogies
• Word analogy
• Letter analogy
• Mixed analogies

However, different types of analogies are used in multiple tests, depending on the kind of ability that you want to quantify in the individual.

Many training tests, both academic and occupational, use numerical analogies to measure competencies in applicants. They are usually presented within the context of logical or abstract reasoning.

How are the premises represented?

According to the operations and characteristics of the premises, we can classify numerical analogies in the following way:

By type of number

They can take into account different numerical sets, the fact of belonging to these sets being the similarity between the premises. Prime, even, odd, integer, rational, irrational, imaginary, natural and real numbers can be sets associated with these types of problems.

1: 3:: 2: 4 The observed analogy is that one and three are the first odd natural numbers. Similarly two and four are the first even natural numbers.

3: 5:: 19: 23 We observe 4 prime numbers where five is the prime number that follows three. Similarly, Twenty-three is the prime number that follows nineteen.

By internal operations of the element

The figures that make up the element can be altered with combined operations, this order of operation being the analogy sought.

231: 6:: 135: 9 The inner operation 2 + 3 + 1 = 6 defines one of the premises. Similarly 1 + 3 + 5 = 9.

721: 8:: 523: 4 The following combination of operations defines the first premise 7 + 2-1 = 8. Checking the combination in the second premise 5 + 2-3 = 4 the analogy is obtained.

By operations of the element with other factors

Multiple factors can act as an analogy between premises through arithmetic operations. Multiplication, division, empowerment and radication are some of the most frequent cases in this type of problem.

2: 8:: 3: 27 It is observed that the third power of the element is the corresponding analogy 2x2x2 = 8 in the same way as 3x3x3 = 27. The relation is x3

5:40:: 7:56 Multiplying the element by eight is the analogy. The ratio is 8x

Applications of numerical analogies

Not only does mathematics find in numerical analogies a highly applicable tool. In fact, many branches such as sociology and biology tend to run into numerical analogies, even in the study of elements other than numbers.

Patterns found in graphs, research and evidence are commonly captured as numerical analogies, facilitating the obtaining and prediction of results. This is still sensitive to faults, because the correct modeling of a numerical structure in accordance with the phenomenon under study is the only guarantor of optimal results.

Sudoku

Sudoku is very popular in recent years due to its implementation in many newspapers and magazines. It consists of a mathematical game where premises of order and form are established.

Each 3 × 3 square must contain the numbers from 1 to 9, preserving the condition of not repeating any value linearly, both vertically and horizontally.

How are the exercises of numerical analogies solved?

The first thing to take into account is the type of operations and characteristics involved in each premise. After finding the similarity, we proceed to operate in the same way for the unknown.

Solved exercises

Exercise 1

10: 2:: 15:?

The first relation that jumps out is that two is a fifth of 10. In this way, the similarity between the premises can be X / 5. Where 15/5 = 3

A possible numerical analogy for this exercise is defined with the expression:

10: 2:: 15: 3

Exercise 2

24 (9) 3

12 (8) 5

32 (?) 6

The operations that verify the first 2 premises are defined: Divide the first number by four and add the third number to that result

(24/4) + 3 = 9

(12/4) + 5 = 8

Then the same algorithm is applied to the row containing the unknown

(32/4) + 6 = 14

Being 24 (9) 3 a possible solution according to the relation (A / 4) + C = B

12 (8) 5

32 (14) 6

Assuming a hypothetical general structure A (B) C in each premise.

In these exercises it is shown how different structures can house the premises.

Exercise 3

26: 32:: 12: 6

14: 42:: 4:?

Form ii) is evidenced to arrange the premises where 26 is a 12 as 32 is a 6

At the same time there are internal operations applicable to the premises:

2 x 6 = 12

3 x 2 = 6

Once this pattern is observed, it is proved in the third premise:

1 x 4 = 4

It only remains to apply this operation once more to obtain the possible solution.

4 x 2 = 8

Obtaining 26: 32:: 12: 6 as a possible numerical analogy.

14: 42:: 4: 8

Proposed exercises to solve

It is important to practice to achieve mastery of these types of problems. As in many other mathematical methods, practice and repetition are essential to optimize resolution times, energy expenditure and fluency in finding possible solutions.

Find the possible solutions to each numerical analogy presented, justify and develop your analysis:

Exercise 1

104: 5:: 273:?

Exercise 2

8 (66) 2

7 (52) 3

3 (?) 1

Exercise 3

10A 5B 15C 10D 20E?

Exercise 4

72: 10:: 36: 6

45: 7::?: 9

References

1. Holyoak, KJ (2012). Analogy and relational reasoning. In KJ Holyoak & RG Morrison. The Oxford handbook of thinking and reasoning New York: Oxford University Press.
2. ANALOGICAL REASONING IN CHILDREN. Usha Goswami, Institute of Child Health, University College London, 30 Guilford St., London WC1N1EH, UK
3. The Arithmetic Teacher, Volume 29. National Council of Teachers of Mathematics, 1981. University of Michigan.
4. Most powerful handbook for reasoning, Shortcuts in reasoning (verbal, non-verbal and analytical) for competitive exams. Disha publication.
5. Learning and teaching number theory: Research in cognition and instruction / edited by Stephen R. Campbell and Rina Zazkis. Ablex publishing 88 Post Road West, Westport CT 06881