Hannah and her sister Rita each had some beads. When Hannah gave 80 beads to Rita, she had 3 times as many beads as Rita. When Hannah gave another 32 beads to Rita, she still had 156 more than Rita. How many beads did Rita have at first?
Question
Hannah and her sister Rita each had some beads. When Hannah gave 80 beads to Rita, she had 3 times as many beads as Rita. When Hannah gave another 32 beads to Rita, she still had 156 more than Rita. How many beads did Rita have at first?
Solution
Let's denote the number of beads Hannah had at first as H and the number of beads Rita had at first as R.
From the problem, we have two equations:
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After Hannah gave 80 beads to Rita, she had 3 times as many beads as Rita. This can be written as: H - 80 = 3 * (R + 80)
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When Hannah gave another 32 beads to Rita, she still had 156 more than Rita. This can be written as: H - 80 - 32 = (R + 80 + 32) + 156
Solving these two equations, we get:
From equation 1) H - 3R = 80 * 3 From equation 2) H - R = 80 + 32 + 156
Solving these two equations, we get R = 112 and H = 416.
So, Rita had 112 beads at first.
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