Kate had 5 times as many beads as Shelly at first. After, Kate gave away 648 beads and Shelly received 216 beads, they had the same number of beads in the end. How many beads did Kate have at first? Show your steps.

Answer 1

Let's assume that Kate had $x$ beads at first.
Since Kate had 5 times as many beads as Shelly, Shelly had $\frac{1}{5}x$ beads at first.
After Kate gave away 648 beads, she had $x - 648$ beads left.
After Shelly received 216 beads, she had $\frac{1}{5}x + 216$ beads.
Since they had the same number of beads in the end, we can set up the equation:
$x - 648 = \frac{1}{5}x + 216$
Multiplying both sides of the equation by 5 to get rid of the fraction, we get:
$5x - 3240 = x + 1080$
Subtracting x from both sides of the equation, we get:
$4x - 3240 = 1080$
Adding 3240 to both sides of the equation, we get:
$4x = 4320$
Dividing both sides of the equation by 4, we get:
$x = 1080$
So, Kate had 1080 beads at first.