# The sum of a two digit number and its reversal is 121 and their difference is 9 what is that number

Suppose, the units place digit of the two digit number is y and the tens place digit is x.∴ the number is 10x + y∴ the number obtained by reversing the digits is 10y + x

∴ from the given conditions,

`(10x+y)+(10y+x)=121`

∴ 11x+11y=121                                ∴ x+y=11 .............(I)

Also, x=y+7                                      ∴ x-y=7 ................(II)

∴ Adding (I) and (II), `2x=18       x=9`

∴ from (I) `a+y=11`     `    y=2`

∴ the two digit number is 29.

Take any 2-digit number. Reverse the digits to make another 2-digit number. Add the two numbers together.

How many answers do you get which are still 2-digit numbers?

What do the answers have in common?

Specific Learning Outcomes

Add 2-digit numbers with and without renaming

Description of Mathematics

This problem practices the addition of 2-digit numbers. Encourage the students to share the methods that they use to solve the problem. For example some students may use place value while others will find it easier to use a rounding method. 91 + 19    place value: 91 + 9 + 10

rounding: 91 + 20 – 1

This problem also offers the opportunity for students to "play" with numbers. As well as practising addition the students are encouraged to look for patterns in their answers. This play encourages students to increase their understanding of numbers and how they relate to one another.  It also helps develop problem solving skill and creativity.

As numbers are 'reversed' they swap places. (eg. 41 to 14) It is therefore important to discuss what is happening to the place value of the numbers.

Required Resource Materials

Take any 2-digit number. Reverse the digits to make another 2-digit number. Add the two numbers together. How many answers do you get which are still 2-digit numbers?

What do the answers have in common?

### Teaching Sequence

1. Introduce the problem – you could do this by writing 2 reversed 2-digit numbers eg 14 and 41. Ask the students what they can tell you about the 2 numbers. If they identify that the digits have swapped places then introduce the problem.
2. It is important that they are clear about how to reverse numbers and that they understand the difference between 2 and 3-digit numbers.
3. You may decide to do an example with the class.
4. Discuss with the students the strategies that they could use to add 2-digit numbers.
5. Let the students work on the problem individually before putting them in small groups. Some students are quicker than others when computing and it is important that all students have the opportunity to "play" with the problem before getting them to share their findings. If all students  have some work to bring to the group they are more likely to be involved in the solution.
6. As you circulate, encourage the students to explain how they are getting the answers.
7. Ask the students how they could organise their reversed numbers so that they could look for patterns in the answers. A good starting point would be to sort the 2 and 3-digit answers into lists or they may decide to identify the reversed numbers that give a 2-digit answer on the hundreds board.
8. Share patterns.

#### Extension to the problem

Is there a pattern in the numbers that give 3-digit sums?

### Solution

There are many patterns that can be found in this problem.  Let's try a few numbers and see what we get: 13 + 31 = 44 26 + 62 = 88 47 + 74 = 121 54 + 45 = 99 68 + 86 = 154 Now we can see that if the sum of the digits in the 2-digit number is less than 10 then the sum of the reversed numbers is less than 100. 27 + 72 = 99 The sum of the digits in the 2-digit number determines the sum of the reversed numbers in the following way: If the sum is 6 the answer is 66 (24 + 42 = 66; 15 + 51 = 66 etc)

If the sum is 8 then the sum of the reversed numbers is 88.

You might support your students to notice that the sum in every case above is a multiple of 11.

#### Solution to the Extension:

Once again the 3-digit sums are all multiples of 11.  To see this notive that 68 + 86 gives the same answer as 66 + 88.  Now both 66 and 88 are multiples of 11, so the sum is too.

The sum of a two digit number and the number obtained by reversing the digits is 88 . If the digits of the number differ by 2 , find the numbers.

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