Use algebra to prove that 0.2083 recurring multiplied by 0.26 recurring is equal to
Let’s begin by converting 0.2083 recurring into a fraction in its simplest form. Notice how the question asks us to use algebra. So it’s not enough to type this into your calculator.
We begin by defining 𝑥 to be 0.2083 recurring. Our aim is to create two decimals with the exact same sequence of digits after the
decimal point. To achieve this, we’re going to start by multiplying our number by 1000. By doing so, we’ve made sure that only the recurring part of this number is after the
Now, we need to find a way to multiply this equation to create another number with
point three recurring after the decimal point. The simplest way to do that is just to multiply all by 10. Multiplying by 10 moves all the digits to the left once. So we should still end up with three recurring after the decimal point. Multiplying by 10 gives us 10000𝑥 is equal to 2083.3 recurring.
Now that we have two numbers with the exact same sequence of digits after the decimal
point, we can subtract the smaller one from the larger. Writing this smaller equation directly below the larger one can make it a little bit
easy to see what’s going on: 10000𝑥 minus 1000𝑥 is 9000𝑥. Then, when we subtract, all the numbers after the decimal point essentially
disappear. We’re left with 1875.
At this stage, we’re just left with a linear equation that we can solve by dividing
everything by 9000, giving us 𝑥 is equal to 1875 over 9000 which we then need to
simplify fully. Now, at this stage, we can just type this into our calculator and it will simplify
the fraction for us. However, you very well might come across to sum like this in a non-calculator
paper. So it’s useful to know where to go next.
Perhaps, the easiest number to stop by dividing by is five. We can do this using the bus stop method. However, if we’re careful, we might be able to spot that they’re also divisible by
25. Twenty-fives into one or 18 doesn’t go. So we’re going to do 25 into 187 which is seven with a remainder 12. Twenty-fives into 125 goes five times. Therefore, 1875 divided by 25 is 75.
Let’s repeat this process by dividing 9000 by 25. Twenty-fives into 90 goes three times with a remainder 15. 25 goes into a 150 six times and obviously zero times into zero. Our fraction initially simplifies into 75 over 360. We can simplify further by dividing both of these by 15 which gives us a fully
simplified fraction of five twenty-fourths. Remember we originally defined 0.2083 recurring as 𝑥. And we’ve now shown that 𝑥 is equal to five twenty-fourths. So that means 0.2083 recurring is the same as five twenty-fourths.
Let’s now do the same for the second recurring decimal. This time though we’ll call it 𝑦 so we don’t get confused. Let 𝑦 be equal to 0.26 recurring. Now, at this occasion, I only need to multiply both sides by 10 to get the recurring
part to be immediately after the decimal point: 10𝑦 is equal to 2.6 recurring. Remember I’m trying to create two numbers with the exact same digits after the
decimal point. So if I now multiply this new equation by 10, I get 100𝑦 is equal to 26.6
Now that I’ve created two numbers with six recurring immediately after the decimal
point, I can subtract the smaller one from the larger. Again, it can be easier to rewrite the equation below the larger equation to make
your life easier: 100𝑦 minus 10𝑦 is 90𝑦 and 26.6 recurring minus 2.6 recurring is
24. We can solve this equation by dividing both sides by 90 which gives us 24 over
90. Again, we can simplify by typing these numbers into our calculator or if we’re in a
non-calculator paper, we could spot that they’re both divisible by six. 0.26 recurring is therefore equal to four fifteenths.
Now that we have each of our recurring decimals in fraction form, we can multiply
them as the question asks us to do. We can cross cancel by dividing through by a common factor of five. Five divided by five is one and 15 divided by five is three and a common factor of
four. Four divided by four is one and 24 divided by four is six. Then, we multiply as normal to get an 18th.
Notice that this is what the question wanted us to show them. So our job is done. 0.2083 recurring multiplied by 0.26 recurring is equal to an 18th.