Operations with decimals
We need to divide 0.25 into 1.03075. Now the first thing you want to do when your divisor, the number that you're dividing into the other number, is a decimal, is to multiply it by 10 enough times so that it becomes a whole number so you can shift the decimal to the right. So every time you multiply something by 10, you're shifting the decimal over to the right once. So in this case, we want to switch it over the right once and twice. So 0.25 times 10 twice is the same thing as 0.25 times 100, and we'll turn the 0.25 into 25. Now if you do that with the divisor, you also have to do that with the dividend, the number that you're dividing into. So we also have to multiply this by 10 twice, or another way of doing it is shift the decimal over to the right twice. So we shift it over once, twice. It will sit right over here. And to see why that makes sense, you just have to realize that this expression right here, this division problem, is the exact same thing as having 1.03075 divided by 0.25. And so we're multiplying the 0.25 by 10 twice. We're essentially multiplying it by 100. Let me do that in a different color. We're multiplying it by 100 in the denominator. This is the divisor. We're multiplying it by 100, so we also have to do the same thing to the numerator, if we don't want to change this expression, if we don't want to change the number. So we also have to multiply that by 100. And when you do that, this becomes 25, and this becomes 103.075. Now let me just rewrite this. Sometimes if you're doing this in a workbook or something, you don't have to rewrite it as long as you remember where the decimal is. But I'm going to rewrite it, just so it's a little bit neater. So we multiplied both the divisor and the dividend by 100. This problem becomes 25 divided into 103.075. These are going to result in the exact same quotient. They're the exact same fraction, if you want to view it that way. We've just multiplied both the numerator and the denominator by 100 to shift the decimal over to the right twice. Now that we've done that, we're ready to divide. So the first thing, we have 25 here, and there's always a little bit of an art to dividing something by a multiple-digit number, so we'll see how well we can do. So 25 does not go into 1. 25 does not go into 10. 25 does go into 103. We know that 4 times 25 is 100, so 25 goes into 100 four times. 4 times 5 is 20. 4 times 2 is 8, plus 2 is 100. We knew that. Four quarters is $1.00. It's 100 cents. And now we subtract. 103 minus 100 is going to be 3, and now we can bring down this 0. So we bring down that 0 there. 25 goes into 30 one time. And if we want, we could immediately put this decimal here. We don't have to wait until the end of the problem. This decimal sits right in that place, so we could always have that decimal sitting right there in our quotient or in our answer. So we were at 25 goes into 30 one time. 1 times 25 is 25, and then we can subtract. 30 minus 25, well, that's just 5. I mean, we can do all this borrowing business, or regrouping. This can become a 10. This becomes a 2. 10 minus 5 is 5. 2 minus 2 is nothing. But anyway, 30 minus 25 is 5. Now we can bring down this 7. 25 goes into 57 two times, right? 25 times 2 is 50. 25 goes into 57 two times. 2 times 25 is 50. And now we subtract again. 57 minus 50 is 7. And now we're almost done. We bring down that 5 right over there. 25 goes into 75 three times. 3 times 25 is 75. 3 times 5 is 15. Regroup the 1. We can ignore that. That was from before. 3 times 2 is 6, plus 1 is 7. So you can see that. And then we subtract, and then we have no remainder. So 25 goes into 103.075 exactly 4.123 times, which makes sense, because 25 goes into 100 about four times. This is a little bit larger than 100, so it's going to be a little bit more than four times. And that's going to be the exact same answer as the number of times that 0.25 goes into 1.03075. This will also be 4.123. So this fraction, or this expression, is the exact same thing as 4.123. And we're done!