Video Transcript
Fill in the blank. If 1.6 times 𝑛 is equal to one, then 𝑛 is equal to what. Give your answer as a fraction.
In this question, we are given an equation involving an unknown 𝑛 and asked to find the value of 𝑛 that satisfies the equation giving our answer as a fraction.
To answer this question, we can begin by looking at the equation that we are given. We can see that we are multiplying two numbers together and the result is equal to one. We can then see that this equation is very similar to the multiplicative inverse property of rational numbers, which tells us that all nonzero rational numbers have a multiplicative inverse. In general, we have that if 𝑎 and 𝑏 are nonzero integers, then 𝑎 over 𝑏 times 𝑏 over 𝑎 is equal to one.
To apply this result to the given equation, we need two things. First, we need to check that 1.6 and 𝑛 are both rational numbers. We know that 1.6 is rational because it has a finite decimal expansion. And we know that 𝑛 is rational since its product with a rational number is rational. Second, we need to rewrite 1.6 as a fraction. We can do this by rewriting 1.6 as 16 over 10. We can then cancel the shared factor of two in the numerator and denominator to obtain eight over five. Substituting this value into the equation gives us eight-fifths times 𝑛 is equal to one.
There are now multiple ways that we can find the value of 𝑛. One way is to directly apply the multiplicative inverse property to the equation to see that 𝑛 must be equal to the reciprocal of eight-fifths, which is five-eighths. Another method is to multiply both sides of the equation by five-eighths. We can then apply the multiplicative inverse property and the multiplicative identity property along with the associativity of multiplication to get that 𝑛 is equal to five-eighths. Finally, we can also note that this is equivalent to dividing both sides of the equation by eight-fifths.
In either case, we have shown that the solution to the equation is 𝑛 is equal to five-eighths.
