Select the values of 𝑎 and 𝑏 which satisfy this statement: 𝑎 and 𝑏 are positive numbers such that 𝑎 is less than one, 𝑏 is less than one, and 𝑎𝑏 is less than one. We have five possible answers. 𝑎 equals 3.97 and 𝑏 equals 6.21, or 𝑎 equals 0.61 and 𝑏 equals 0.17, or 𝑎 equals 0.36 and 𝑏 equals 1.1, or 𝑎 equals 4.21 and 𝑏 equals 0.43, or finally 𝑎 equals 5.66 and 𝑏 equals 3.01.
We know that the part of math that we call algebra involves using letters to represent numbers. And this problem is all about trying to work out what the values of the letter 𝑎 and the letter 𝑏 are. And the statement in the question gives us some clues. These are going to help us to work out what the value of 𝑎 and 𝑏 are going to be. The first clue we’re given is that 𝑎 and 𝑏 are positive numbers. In other words, these are numbers that are greater than zero. They’re numbers that appear to the right of zero on the number line.
If we quickly glanced down our possible answers, there are no negative signs. We can’t see any negative numbers. So all of our possible answers include positive numbers. So what are the clues that we’re given? The statement tells us that 𝑎 and 𝑏 are positive numbers such that, firstly, 𝑎 is less than one. Remember that the wider part of this symbol always points to the larger value. So when we draw the arrow pointing this way, it means is less than. So let’s go through our possible answers and see which ones show the value 𝑎 being less than one.
In the first pair of values, 𝑎 is worth 3.97. This isn’t less than one. This is more than one. So we can see that the first pair of values is not going to be correct. We can cross this one off. In our second pair of numbers, 𝑎 has a value of 0.61 and 0.61 is less than one. This could be our answer. Our third pair of numbers has 𝑎 having a value of 0.36. Again, this is less than one. So this could be our answer as well. In our fourth answer, 𝑎 has a value of 4.21 and this is greater than one.
So we can cross through this as a possible answer. And we can also cross through our last pair of numbers because 𝑎 has a value of 5.66 here, which again is greater than one, not less than one. So there are only two possible answers where 𝑎 has a value less than one. Which of these could be the correct answer? We need to continue to read the clues in the question. So 𝑎 and 𝑏 are positive numbers such that 𝑎 is less than one and then 𝑏 is less than one too. Look how the symbol is the same. This time, it won’t take so long to go through. We’ve only got two possible answers to look at.
The first pair of values 𝑏 has a value of 0.17 and this is less than one. So this could be our answer. Now, if we look at our second pair of numbers. Although 𝑎 was less than one, we can see that 𝑏 has a value of 1.1. That’s one and one 10th, which is greater than one. So unfortunately, we’re going to have to cross through this. This is not our answer. There’s only one pair of numbers such that 𝑎 is less than one and 𝑏 is less than one. So it looks like we found our answer.
But there’s one more clue in the question and we need to make sure that this is true too. 𝑎 is less than one, 𝑏 is less than one, and also 𝑎𝑏 is less than one. And when we’re working with letters in algebra, if we write two letters next to each other or a letter next to a number, it means that we need to multiply those two values together. So when we say 𝑎𝑏 is less than one, what we really mean is 𝑎 multiplied by 𝑏 is less than one. So to check whether this is true, we need to see whether 0.61 multiplied by 0.17 is less than one.
To do this, we can use a form of vertical multiplication. To begin with, we can write both numbers vertically: 0.61 multiplied by 0.17. Now, it would be a lot easier if we were multiplying whole numbers now and not decimals. And there is a way that we can do this. First, we make a note of how many decimal places each number has. 0.61 has two decimal places and 0.17 also has two decimal places. The next thing we can do is to remove the decimal point in the zeroes and the ones places. And we’re left with 61 multiplied by 17.
Now, what we’ve done is we’ve removed two decimal places from the first number and two decimal places from the second number. So as long as we put four decimal places back in at the end, we found the correct answer. To begin with, we can multiply both digits in 61 by seven. One times seven equals seven. Six sevens are 42. So six 10s multiplied by seven equals 42 10s or 420. Next, we need to multiply both digits in 61 by 10. This is something we can do really quickly because we know that the digits to shift one place to the left. 61 multiplied by 10 is 610. And now, we can add to find the answer.
Seven ones plus zero equals seven. Two 10s plus one 10 equals three 10s. And four hundreds plus six hundreds equals 10 hundreds or 1000. So if we know that 61 multiplied by 17 is 1037, as we’ve just said, the answer to 0.61 multiplied by 0.17, we’ll have an answer with four decimal places: one two, three, four. The answer is 0.1037.
So is 𝑎 multiplied by 𝑏 less than one? Yes, it is. It’s 0.1037. We’ve checked that all three clues are correct. 𝑎 is less than one, 𝑏 is less than one, and the product of 𝑎 and 𝑏 is also less than one. There’s only one pair of numbers that fits these clues: 𝑎 is 0.61 and 𝑏 equals 0.17.