Video Transcript
Find 𝑛 in the equation
three-fifths times six-sevenths multiplied by four-ninths is equal to three-fifths
multiplied by 𝑛 times four-ninths.
In this question, we are asked to
find the value of 𝑛 that solves the equation. And there are a few different ways
that we could do this. For instance, we could evaluate the
left-hand side of the equation and rearrange to isolate 𝑛 on the right-hand side of
the equation. This method would work and would
give us the correct answer.
However, there is an easier method
that we can use by comparing both sides of the equation. We see that both sides of the
equation involve products with the same numbers evaluated in a different order. We can note that this is very
similar to the associative property of the multiplication of rational numbers, which
tells us that if 𝑎, 𝑏, and 𝑐 are rational numbers, then 𝑎 times 𝑏 multiplied by
𝑐 is equal to 𝑎 multiplied by 𝑏 times 𝑐. In other words, we can evaluate the
product of three rational numbers in any order.
It is worth noting that we can
conclude that 𝑛 is rational by using the closure properties of the multiplication
of rational numbers. However, we will apply the
associative property to the left-hand side of the equation for simplicity. Applying the associative property
of the multiplication of rational numbers to the left-hand side of the equation
gives us that three-fifths multiplied six-sevenths times four-ninths is equal to
three-fifths multiplied by 𝑛 times four-ninths.
We can then see that both sides of
the equation share a factor of three-fifths. So we can divide both sides of the
equation by three-fifths in order to simplify. This gives us that six-sevenths
times four-ninths is equal to 𝑛 times four-ninths.
We can now divide both sides of the
equation by four-ninths to solve for 𝑛. It is worth noting that this is the
same as multiplying both sides of the equation by the multiplicative inverse of
four-ninths. We get that 𝑛 is equal to
six-sevenths, which is our final answer.
