5 TWO-DIGIT DIVISIONS RESOLVED - MATHS - 2025

To perform two-digit divisions, you need to know how to divide by single-digit numbers. Divisions are the fourth mathematical operation taught to children in elementary school.

Teaching begins with single-digit divisions - that is, with single-digit numbers - and progresses to divisions between numbers with several digits.

The division process consists of a dividend and a divisor, such that the dividend is greater than or equal to the divisor.

The idea is to get a natural number called a quotient. When multiplying the quotient by the divisor, the result must equal the dividend. In this case, the result of the division is the quotient.

One digit division

Let D be the dividend and d be the divisor, such that D≥dyd is a single-digit number.

The division process consists of:

1. - Choose digits of D, from left to right, until these digits form a number greater than or equal to d.
2. - Find a natural number (from 1 to 9), such that when multiplying it by d the result is less than or equal to the number formed in the previous step.
3. - Subtract the number found in step 1 minus the result of multiplying the number found in step 2 by d.
4. - If the result obtained is greater than or equal to d, then the number chosen in step 2 must be changed to a greater one, until the result is a number less than d.
5. - If not all the digits of D were chosen in step 1, then the first digit from left to right that was not chosen is taken, it is added to the result obtained in the previous step and steps 2, 3 and 4 are repeated.

This process is carried out until the digits of the number D are finished. The result of the division will be the number that is formed in step 2.

Examples of one-digit divisions

To illustrate the steps described above, we will proceed to divide 32 by 2.

- From the number 32, only 3 is taken, since 3 ≥ 2.

- We choose 1, since 2 * 1 = 2 ≤ 3. Note that 2 * 2 = 4 ≥ 3.

- 3 - 2 = 1 is subtracted. Note that 1 ≤ 2, which indicates that the division has been well done so far.

- The digit 2 of 32 is chosen. When it is joined with the result of the previous step, the number 12 is formed.

Now it is as if the division begins again: we proceed to divide 12 by 2.

- Both figures are chosen, that is, 12 is chosen.

- 6 is chosen, since 2 * 6 = 12 ≤ 12.

- Subtracting 12-12 results in 0, which is less than 2.

As the digits of 32 are over, it is concluded that the result of the division between 32 and 2 is the number formed by the digits 1 and 6 in that order, that is, the number 16.

In conclusion, 32 ÷ 2 = 16.

Two-digit divisions

Two-digit divisions are performed similarly to one-digit divisions. With the help of the following examples the method is illustrated.

Examples

First division

It will divide 36 by 12.

- Both figures of 36 are chosen, since 36 ≥ 12.

- Find a number that, when multiplied by 12, the result is close to 36. You can make a short list: 12 * 1 = 12, 12 * 2 = 24, 12 * 3 = 36, 12 * 4 = 48. By choosing 4, the result exceeded 36, therefore, 3 is chosen.

- Subtracting 36-12 * 3 gives 0.

- All the digits of the dividend have already been used.

The result of dividing 36 ÷ 12 is 3.

Second division

Divide 96 by 24.

- Both numbers of 96 must be chosen.

- After investigating it can be seen that 4 must be chosen, since 4 * 24 = 96 and 5 * 24 = 120.

- Subtracting 96-96 gives 0.

- All 96 figures have already been used.

The result of 96 ÷ 24 is 4.

Third d

Divide 120 by 10.

- The first two digits of 120 are chosen; that is, 12, since 12 ≥ 10.

- You must take 1, since 10 * 1 = 10 and 10 * 2 = 20.

- By subtracting 12-10 * 1 you get 2.

- Now the previous result is joined with the third figure of 120, that is, 2 with 0. Therefore the number 20 is formed.

- A number is chosen that when multiplied by 10 is close to 20. This number must be 2.

- Subtracting 20-10 * 2 gives 0.

- All the figures of 120 have already been used.

In conclusion, 120 ÷ 10 = 12.

Fourth d

Divide 465 by 15.

- 46 is chosen.

- After making the list, it can be concluded that 3 must be chosen, since 3 * 15 = 45.

- 46-45 is subtracted and 1 is obtained.

- By joining 1 with 5 (third digit of 465), you get 45.

- 1 is chosen, since 1 * 45 = 45.

- 45-45 is subtracted and 0 is obtained.

- All 465 figures have already been used.

Therefore, 465 ÷ 15 = 31.

Fifth division

Divide 828 by 36.

- Choose 82 (only the first two digits).

- Take 2, since 36 * 2 = 72 and 36 * 3 = 108.

- Subtract 82 minus 2 * 36 = 72 and get 10.

- By joining 10 with 8 (third digit of 828) the number 108 is formed.

- Thanks to step two we can know that 36 * 3 = 108, therefore 3 is chosen.

- By subtracting 108 minus 108 you get 0.

- All 828 figures have already been used.

Finally, it is concluded that 828 ÷ 36 = 23.

Observation

In the previous divisions the final subtraction always resulted in 0, but this is not always the case. This happened because the divisions raised were exact.

When the division is not exact, decimal numbers appear, which must be learned in detail.

If the dividend has more than 3 digits, the division process is the same.

References

1. Barrantes, H., Díaz, P., Murillo, M., & Soto, A. (1988). Introduction to Number Theory. San José: EUNED.
2. Eisenbud, D. (2013). Commutative Algebra: with a View Toward Algebraic Geometry (llustrated ed.). Springer Science & Business Media.
3. Johnston, W., & McAllister, A. (2009). A Transition to Advanced Mathematics: A Survey Course. Oxford University Press.
4. Penner, RC (1999). Discrete Mathematics: Proof Techniques and Mathematical Structures (illustrated, reprint ed.). World Scientific.
5. Sigler, LE (1981). Algebra. Reverte.
6. Zaragoza, AC (2009). Number Theory. Vision Books.