# Rounding off to correct significant figure: If we add 3.46 and 2.3 we get 5.76....

#### Indranil

##### Junior Member
If we add 3.46 and 2.3 we get 5.76.
we can round it to two significant numbers 5.8 because there is the least two significant digit 2.3 but If we add 2.3 X 10^4 + 1.6 X 10^3 = 2.3 X 10^4 + .16 X 10^4 = 2.46 X 10^4. Here can we round off to two significant figures (2.5 X 10^4) because there is also the least two significant digit 2.3 present? Please explain.

#### Jomo

##### Elite Member
If we add 3.46 and 2.3 we get 5.76.
we can round it to two significant numbers 5.8 because there is the least two significant digit 2.3 but If we add 2.3 X 10^4 + 1.6 X 10^3 = 2.3 X 10^4 + .16 X 10^4 = 2.46 X 10^4. Here can we round off to two significant figures (2.5 X 10^4) because there is also the least two significant digit 2.3 present? Please explain.
What are you not understanding? It seems all correct so I assume that you copied it from somewhere and need help understanding it?If you would be more precise in stating which part you are not getting we can help you better.

#### Dr.Peterson

##### Elite Member
If we add 3.46 and 2.3 we get 5.76.
we can round it to two significant numbers 5.8 because there is the least two significant digit 2.3 but If we add 2.3 X 10^4 + 1.6 X 10^3 = 2.3 X 10^4 + .16 X 10^4 = 2.46 X 10^4. Here can we round off to two significant figures (2.5 X 10^4) because there is also the least two significant digit 2.3 present? Please explain.
The only issue I have here is that we only use significant digits to make this decision about rounding when we multiply (or divide). In adding, we use individual decimal places instead. For example, if we add 1.4375 + 243.125, we would round the answer, 244.5625, to the thousandths, 244.563, not to 5 significant digits, 244.56; the reason is that both have known digits in the thousandths place.

You have the right answers, but the wrong reasons.

#### Indranil

##### Junior Member
The only issue I have here is that we only use significant digits to make this decision about rounding when we multiply (or divide). In adding, we use individual decimal places instead. For example, if we add 1.4375 + 243.125, we would round the answer, 244.5625, to the thousandths, 244.563, not to 5 significant digits, 244.56; the reason is that both have known digits in the thousandths place.

You have the right answers, but the wrong reasons.
What do you mean by 'both have known digits in the thousandths place.'? Kindly explain, please

#### Indranil

##### Junior Member
What are you not understanding? It seems all correct so I assume that you copied it from somewhere and need help understanding it?If you would be more precise in stating which part you are not getting we can help you better.
If we add 2.3 X 10^4 + 1.6 X 10^3 = 2.3 X 10^4 + .16 X 10^4 = 2.46 X 10^4. If I round it off, then what will be the answer and why?

#### Subhotosh Khan

##### Super Moderator
Staff member
If we add 2.3 X 10^4 + 1.6 X 10^3 = 2.3 X 10^4 + .16 X 10^4 = 2.46 X 10^4. If I round it off, then what will be the answer and why?
Rounding-off is an economical decision which depends on the application.
You may want to carry a large number of significant digits while designing a Hadron collider or a space-shuttle - high risk endeavor. everything there needs to match within few angstorms.

However, while you are designing a "beer barrel" - a low risk endeavor - you need to match things within 1/16 of an inch. You can always tighten up the straps as needed. So to answer your question in engineering sense:

If we add 2.3 X 10^4 + 1.6 X 10^3 = 2.3 X 10^4 + .16 X 10^4 = 2.46 X 10^4. If I round it off, then what will be the answer and why?

You were given your data with two significant digits. There is a difference between 2.30 * 10^4 and 2.3 * 10^4.

So your answer should be given with two significant digits:

2.3 X 10^4 + 1.6 X 10^3 = 2.3 X 10^4 + .16 X 10^4 = 2.46 X 10^4 = 2.5 * 10^4

There will be significant cost difference between two machined rods of lengths 2.5" and 2.46".

The rod with length 2.46" will cost more because you are demanding more "accurate" machining - hence more machine time and more "unacceptable" product from production line.

Since original data was given with two significant digits - you cannot expect an answer better than two significant digits. It is very nice to be able to write pi = 3.141592654 but for most of our work pi = 22/7 works fine (it took us to the moon and back - safely).

#### Dr.Peterson

##### Elite Member
What do you mean by 'both have known digits in the thousandths place.'? Kindly explain, please
The least significant digit in 1.4375 is the 5 in the ten-thousandths place; in 243.125, it is the 5 in the thousandths place. Here we don't have a ten-thousandths digit, so we can't claim to know the ten-thousandths digit in the sum. But we do know the thousandths place in both numbers, so that is accurate in the sum.

That is, in adding these, we are really adding 1.4375 + 243.1250, but the added zero is not known to be accurate, so we can't use the result in that place. We can use the next place over.

See here and here and here, among many others.

#### Indranil

##### Junior Member
The least significant digit in 1.4375 is the 5 in the ten-thousandths place; in 243.125, it is the 5 in the thousandths place. Here we don't have a ten-thousandths digit, so we can't claim to know the ten-thousandths digit in the sum. But we do know the thousandths place in both numbers, so that is accurate in the sum.

That is, in adding these, we are really adding 1.4375 + 243.1250, but the added zero is not known to be accurate, so we can't use the result in that place. We can use the next place over.

See here and here and here, among many others.

• [*=left][h=4]Simplify, rounding [FONT=&quot]1247 + 134.4 + 450 + 78[/FONT] to the appropriate number of significant digits.[/h]
[FONT=&quot]Looking at each of the numbers they've given me, I see that I will have to round the final answer to the nearest tens place, because [FONT=&quot]450 is only accurate to the tens place. (The other numbers are accurate to the ones, tenths, and ones places, respectively.)[/FONT]
First, I add in the usual way:
[FONT=&quot]1247 + 134.4 + 450 + 78 = 1909.4[/FONT]

...and then I round my result to the tens place, rounding the [FONT=&quot]0 up to [FONT=&quot]1[/FONT] because of the [FONT=&quot]9[/FONT] in the ones place:[/FONT]
[FONT=&quot]1247 + 134.4 + 450 + 78 = 1910
From the above solution, I don't understand the terms below.
[/FONT]

[/FONT]

1. '[FONT=&quot]450[/FONT] is only accurate to the tens place' what does it mean here?
2. '
The other numbers are accurate to the ones, tenths, and ones places, respectively.' what does it mean here?
3. '
I round my result to the tens place, rounding the [FONT=&quot]0[/FONT] up to [FONT=&quot]1[/FONT] because of the [FONT=&quot]9[/FONT] in the ones place' what does it mean here?

#### Dr.Peterson

##### Elite Member

• [*=left]Simplify, rounding 1247 + 134.4 + 450 + 78 to the appropriate number of significant digits.
Looking at each of the numbers they've given me, I see that I will have to round the final answer to the nearest tens place, because 450 is only accurate to the tens place. (The other numbers are accurate to the ones, tenths, and ones places, respectively.)
First, I add in the usual way:
1247 + 134.4 + 450 + 78 = 1909.4

...and then I round my result to the tens place, rounding the 0 up to 1 because of the 9 in the ones place:
1247 + 134.4 + 450 + 78 = 1910
From the above solution, I don't understand the terms below.

1. '450 is only accurate to the tens place' what does it mean here?
2. '
The other numbers are accurate to the ones, tenths, and ones places, respectively.' what does it mean here?
3. '
I round my result to the tens place, rounding the 0 up to 1 because of the 9 in the ones place' what does it mean here?
You're quoting from the first link I gave, right?

The author takes 450 as if it were 4.5*10^2, with the zero not known (because it might have been rounded to the nearest ten). I would say that is ambiguous. But if we make that assumption, then the tens digit (5) is accurate, but the ones digit (0) may not be.

"Accurate to the x place" means that the x place is the rightmost digit that is considered accurate. Apart from that zero, it is just the last digit in the number.

The sum comes to 1909.4; since the ones and tenths places can't be trusted, we round to the tens place, in the usual way: drop the digits to the right of it (1900), but then add 1 in the tens place because the first dropped digit, 9, is 5 or more (1910).

#### Indranil

##### Junior Member
You're quoting from the first link I gave, right?

The author takes 450 as if it were 4.5*10^2, with the zero not known (because it might have been rounded to the nearest ten). I would say that is ambiguous. But if we make that assumption, then the tens digit (5) is accurate, but the ones digit (0) may not be.

"Accurate to the x place" means that the x place is the rightmost digit that is considered accurate. Apart from that zero, it is just the last digit in the number.

The sum comes to 1909.4; since the ones and tenths places can't be trusted, we round to the tens place, in the usual way: drop the digits to the right of it (1900), but then add 1 in the tens place because the first dropped digit, 9, is 5 or more (1910).
1. Why 'the ones and tenths places can't be trusted'?
2. Where did you get 1900 from ' drop the digits to the right of it (1900)'?
3. Why is it '1910', it should be '191' because 5 or more than 5 becomes 1 and is added to the previous digit? Please clear I am stuck

#### Dr.Peterson

##### Elite Member
1. Why 'the ones and tenths places can't be trusted'?
2. Where did you get 1900 from ' drop the digits to the right of it (1900)'?
3. Why is it '1910', it should be '191' because 5 or more than 5 becomes 1 and is added to the previous digit? Please clear I am stuck
Perhaps we need to back up. What DO you understand about significant figures and rounding?

Do you know that when we use rounded numbers or measurements, digits past the last one are considered not to be known/accurate? Then, if we add numbers together, adding a known digit to an unknown one (not present in the measured value) we "can't trust it" -- we don't know its value. We can't just call it zero; it might be anything.

Do you understand how to round a number to the nearest ten? Show me how you would do that for 4567.

#### Indranil

##### Junior Member
Perhaps we need to back up. What DO you understand about significant figures and rounding?

Do you know that when we use rounded numbers or measurements, digits past the last one are considered not to be known/accurate? Then, if we add numbers together, adding a known digit to an unknown one (not present in the measured value) we "can't trust it" -- we don't know its value. We can't just call it zero; it might be anything.

Do you understand how to round a number to the nearest ten? Show me how you would do that for 4567.
If I round off 4567, it should be 4570.

#### Dr.Peterson

##### Elite Member
If I round off 4567, it should be 4570.
Correct. You dropped (that is, changed to zero) the ones place, making 4560, and then added 1 to the tens place because of the 7: 4560 + 10 = 4570.

So how do you get 191 when you round 1909.4?

#### Indranil

##### Junior Member
Correct. You dropped (that is, changed to zero) the ones place, making 4560, and then added 1 to the tens place because of the 7: 4560 + 10 = 4570.

So how do you get 191 when you round 1909.4?
If I round it off to the nearest ten, I get 1,910. But how can I get 191?

#### Dr.Peterson

##### Elite Member
1. Why 'the ones and tenths places can't be trusted'?
2. Where did you get 1900 from ' drop the digits to the right of it (1900)'?
3. Why is it '1910', it should be '191' because 5 or more than 5 becomes 1 and is added to the previous digit? Please clear I am stuck
If I round it off to the nearest ten, I get 1,910. But how can I get 191?
That's what I'm trying to figure out -- you are the one who said it!

#### Indranil

##### Junior Member
That's what I'm trying to figure out -- you are the one who said it!
Thank you for your kind efforts. Thanks a lot.

#### stapel

##### Super Moderator
Staff member
website said:
First, I add in the usual way:
1247 + 134.4 + 450 + 78 = 1909.4

...and then I round my result to the tens place, rounding the 0 up to 1 because of the 9 in the ones place:
1247 + 134.4 + 450 + 78 = 1910
Why is it '1910', it should be '191'...
Why would a value that's almost two thousand be "rounded" (??) down to less than two hundred? Rounding involves editing place values, not removing significant digits entirely! 