Anthony knows how to add and subtract zero, 0.25, 0.5, 0.75, and one. Look at how he uses these numbers to help him estimate sums of decimal numbers. By rounding to the nearest multiple of 0.25, he finds that 1.31 plus 0.42 is about 1.75. Estimate the sum by rounding each number to the nearest multiple of 0.25. 2.17 plus 1.89 equals what. And then, estimate the difference by rounding each number to the nearest multiple of 0.25. 4.69 subtract 0.42 equals what.
There’s a lot to read and take in in this problem. So let’s go through it step by step. Firstly, we’re told that Anthony knows how to add and subtract five numbers. These are zero, 0.25, 0.5, 0.75, and one. These are all multiples of twenty-five hundredths or 0.25. And because Anthony finds it easy to add and subtract using these numbers, he uses them to help him estimate sums and differences of decimal numbers. And we’re given a diagram to show us exactly what he does. And it’s worth taking a moment just to look at it to try to understand what’s happening.
Firstly, Anthony has a calculation to work out. He needs to find the sum of 1.31 plus 0.42. And we can see this vertical calculation on the left. And we can imagine Anthony saying to himself, I don’t want to find out the exact amount. I just need to find an estimate. So what I’ll do is round both numbers to the nearest multiple of 0.25 because I know how to add and subtract those. And so, this is what he does. And he uses a number line to help. Now, this number line is interesting. Firstly, we can see that all the multiples of 0.25 have been marked. They each have a slightly longer line than the other lines on there. And of course, we can see the labels there as well. So these multiples of 0.25 are very clear.
But as well as the longer notches, we can also see there are some smaller marks on the number line too. If we look at where these are placed, we can see that these represent tenths. So, for example, if we’re looking for where 0.81 is going to be, we can find eight-tenths and use this to help us. So these small notches are very useful. The first number in Anthony’s calculation is 1.31. So we know Anthony must locate 1.31 on his number line, which is where the pink circle is. And then, he looks for the nearest multiple of 0.25 that he can round this to. We can see that the pink circle is in between 1.25 and also 1.5. And it’s nearest to 1.25 And so, the first number in Anthony’s estimate is going to be 1.25.
The second number in Anthony’s addition is 0.42. So once again, he locates this on a number line. And in this time, he marks it with a blue circle. This time, the multiples of 0.25 that the blue circle’s in between are 0.25 itself and 0.5. And 0.42 is nearest to 0.5. So Anthony uses this as the second number in his estimate. And because he knows how to add these numbers quickly, he knows that 1.25 plus 0.5 equals 1.75. And this number is his estimate. By rounding to the nearest multiple of 0.25, he finds that 1.31 plus 0.42, that’s his original calculation, is about 1.75.
Now, we’re asked to solve two problems of our own. To give us space to do this, let’s remove Anthony’s example. Our first calculation is an addition. We need to estimate the sum by rounding each number to the nearest multiple of 0.25. Again, we can sketch a number line, just like Anthony’s, to help us. The multiples of 0.25 are using larger notches and the smaller notches represent tenths. Now, we can round each of our decimals to the nearest multiple of 0.25. Firstly, 2.17. If this is where two is on our number line, then this small notch will be two and one-tenth or 2.1. And this second one must be 2.2. So we can say that 2.17 will be around about here on the number line. Which multiple of 0.25 is this nearest to? It’s nearest to 2.25. So the first number in our estimated calculation is going to be 2.25.
But what number are we going to add to it? Let’s locate 1.89 in our number line. Here’s where 1.8 will be. And the second arrow represents 1.9. 1.89 is very, very close to 1.9, so we can draw this dot here. And we can see that the multiple of 0.25 that this is nearest to is the number two. So the second number in our estimation that we need to add is two. Notice how we write the digit two in the ones place. Now, just like Anthony did, we can find the estimate by adding these numbers quickly. 2.25 plus two equals 4.25. Our estimation for the answer to 2.17 plus 1.89 is 4.25.
In the second calculation, we need to find the difference between two numbers. But we still need to use the same rounding method. The first number in our subtraction is 4.69. 4.69 is only one hundredth away from 4.7. If we locate 4.7 on our number line, we can draw a circle. So we can say that the multiple of 0.25 that 4.69 is nearest to is 4.75. The first number in our estimation is going to be 4.75. But what number do we need to subtract?
This time, we need to locate 0.42 on a number line. 0.42 is two hundredths greater than 0.4. So if we find where 0.4 is in our number line, it will be just past that, perhaps about here. We can see that the nearest multiple of 0.25 to 0.42 is 0.5. We round the number up. The second number in our subtraction is 0.5. Now, we can find the difference by subtracting our two estimated numbers. In the hundredths place, we have five. And in our second number, we don’t have any hundredths. So we could actually put zero there as a placeholder. Five take away zero equals five. Seven-tenths take away five-tenths leaves us with two-tenths.
And before we subtract the ones digits, we need to remember we’re working with decimals here. So we need to include a decimal point in our answer. Four ones take away zero ones equals four. In this problem, we’ve estimated the sum and the difference of pairs of decimal numbers by rounding them to the nearest multiple of 0.25. Our estimate for the sum of 2.17 and 1.89 is 4.25. And our estimate for the difference between 4.69 and 0.42 is also 4.25.