Five-eighths of a number is 20. Find ten-quarters of the number.
Let’s give that number a name. We can call it 𝑥 since we don’t currently know the value of that number. We do know, however, that five-eighths of this number is 20. So we can write five-eighths of 𝑥 is equal to 20. Remember, in maths, of just means to multiply. So we can write this as five-eighths multiplied by 𝑥 equals 20. In algebra, we try to avoid using a multiplication symbol and simply write five-eighths 𝑥 is equal to 20.
Once we have this, there are two methods we can use to solve this problem. We could first solve the equation to find the value of the original number. To solve, we can start by multiplying both sides of the equation by eight. That leaves us with five 𝑥 is equal to 160. We can do 20 multiplied by eight in our head really quickly. Two times eight is 16. And 20 is 10 times larger than two. So 20 multiplied by eight must be 160.
A common mistake here is to think that you need to multiply the top of this fraction by eight. In fact, since five is being divided by eight, when you multiply it by eight, it simply cancels that out and leaves us with five. Next, we can divide both sides of this equation by five. Now, we could use the bus stop method to divide 160 by five. But another way is to divide it by 10 and then double the answer. 160 divided by 10 is 16. And 16 multiplied by two is 32. So 160 divided by five is 32. And we’ve calculated the original number.
Next, we need to calculate ten-quarters of this number. To do this, we first find one quarter by dividing it by four. 32 divided by four is eight. Since we’re looking to find the value of ten-quarters, we simply multiply this by 10. Eight multiplied by 10 is 80. So ten-quarters of the number is 80.
Alternatively, we can use scaling to work out the answer to this. We can multiply both sides of this equation by two. And that gives us a value of ten-eighths of 𝑥 is equal to 40. Remember, multiplying by two is just like multiplying by two over one. Five multiplied by two is 10. And eight multiplied by one is eight.
Next, we simplify this fraction by dividing both the numerator and the denominator by two. That gives us five-quarters of 𝑥 is equal to 40. Then, we can multiply by two again to get the value of ten-quarters of 𝑥 as being 80. Once again, we’ve shown that ten-quarters of the number is 80.