Convert Digits to Number

If the problem involves interchanging of the digits in the integer then you would need to convert from the digits to numbers and vice versa. To convert the digits to numbers, we need to multiply with the digit with the place value of the digit.

For example, the value of the number formed by the digit 4 in the ten’s place and the digit 3 in the one’s place is 4 × 10 + 3 × 1

This type of digit problems is shown in the following example

Interchanging Of Digits Problems

Example:

The sum of the digits of a two-digit number is 11. If we interchange the digits then the new number formed is 45 less than the original. Find the original number.

Solution:

Step 1: Assign variables :

Let x = one’s digit

t = ten’s digit

Sentence: The sum of the digits of a two-digit number is 11.

x + t = 11

Isolate variable x

x = 11 – t (equation 1)

Step 2: Convert digits to number

Original number = t × 10 + x

Interchanged number = x × 10 + t

Sentence: If we interchange the digits then the new number formed is 45 less than the original.

Interchanged = Original – 45

x × 10 + t = t × 10 + x – 45

10x + t = 10t + x – 45

10x – x + t = 10t – 45 (–x to both sides)

10x – x = 10t – t – 45 (– t to both sides)

10x – x + 45 = 10t – t (+ 45 to both sides)

10t – t = 10x – x + 45 (Rewrite equation with t on the left hand side)

Combine like terms

10t – t = 10x – x + 45

9t = 9x + 45 (equation 2)

Substitute equation 1 into equation 2

9t = 9(11 – t) + 45

9t = 99 – 9t + 45

Isolate variable t

9t + 9t = 99 + 45

18t = 144

t=144/18=8

The ten’s digit is 8. The one’s digit is 11 – 8 = 3

Answer: The number is 83.

MATH 2-11 | CUENCO FREDDIEAN R.

Numbers, Numerals and Digits

Number

number 5

A number is a count or measurement, that is really an idea in our minds.

We write or talk about numbers using numerals such as "5" or "five".

But we could also hold up 5 fingers, or tap the table 5 times.

These are all different ways of referring to the same number.

There are also special numbers (like π (Pi)) that can't be written exactly, but are still numbers because we know the idea behind them.

Numeral

A numeral is a symbol or name that stands for a number.

Examples: 3, 49 and twelve are all numerals.

So the number is an idea, the numeral is how we write it.

Digit

A digit is a single symbol used to make numerals.

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are the ten digits we use in everyday numbers.

numeral and digits

Example: the numeral 153 is made up of 3 digits ("1", "5" and "3").

Example: the numeral 46 is made up of 2 digits ("4", and "6").

Example: the numeral 9 is made up of 1 digit ("9"). So a single digit can also be a numeral

Digits -> Numerals -> Numbers

So digits make up numerals, and numerals stand for an idea of a number.

Just like letters make up words, and words stand for an idea of the thing

sghfasdf

talking

Number Instead of Numeral

But often people (including myself) say "Number" when they really should say "Numeral" ... it doesn't really matter if you do that, because other people understand you.

But try to use "digit" only when talking about the single symbols that make up numerals, OK?

Other Types of Digits and Numerals

We are all used to using numerals like "237" and "99", but the Romans used Roman Numerals, and there have been many other digits and numerals used throughout history.

Question 122157: A two-digit number is 11 times its units digit. The sum of the digits is 12. Find the number.

To work this problem, you have to think a little bit about what numbers mean.

You are told that you have a two-digit number. That means the number contains some tens (call

the number of tens T) and some units (call the number of units U).

This means that the unknown number is written as TU.

The last part of the problem tells you that the sum of the digits in the number is 12. This

MATH 2-11 | CUENCO FREDDIEAN R.

means that T + U = 12.

Since T is the number of tens in the number and U is the number of units in the number, the

value of the number is T times ten plus U times 1 .... or just 10T + U. [For example,

the number 23 contains 2 tens plus 3 units.]

So again, the number consists of 10T + U and the problem tells you that this is equal to

11 times U. In equation form this is written as:

10T + U = 11U

Now you have two equations with two unknowns:

T + U = 12

10T + U = 11U

Let's get the bottom equation in standard form by getting the two terms with the variables

all on one side of the equation. Do that by subtraction 11U from both sides of the equation.

When you do that subtraction the bottom equation becomes:

10T - 10U = 0

and the two equations that we have are now:

T + U = 12

10T - 10U = 0

Let's plan to solve this by variable elimination. One way we can do that is to multiply

the entire top equation (both sides and all terms) by 10. This will make the top equation

become 10T + 10U = 120 and the pair of equations is then:

10T + 10U = 120

10T - 10U = 0

If we then add the two equations vertically in columns, notice that the +10U and the -10U

cancel each other out ... so the terms containing U are gone. The vertical addition results

in the equation:

20T = 120

You can solve this for T by dividing both sides of this equation by 20 to get:

.

T = 120/20 = 6

This tells us that the number we are looking for contains 6 tens ... so it is in the sixties.

We can return to the equation T + U = 12 and since we now know that T is 6, this equation

becomes 6 + U = 12. You can solve this equation for U by subtracting 6 from both sides to

get rid of the 6 on the left side. When you do that subtraction, the result is U = 6.

So now we know that the number TU is 66.

Let's check it to see that it satisfies the original problem:

MATH 2-11 | CUENCO FREDDIEAN R.

Is the number equal to 11 times its units number? That means, does the number 66 equal 11 times

the units number of 6. Yes, 11 times 6 is equal to 66.

Does the sum of its digits equal 12? Yes, 6 + 6 does equal 12.

So we have checked the problem and we know for certain that the two-digit number we were asked to

find is 66.

Hope this helps you to understand the problem and a method that can be used for solving it.

MATH 2-11 | CUENCO FREDDIEAN R.