Nora and Renata had an equal number of polished stones at first. After Nora received 65 more stones and Renata lost 23 stones, Nora had three times as many stones as Renata. How many stones did they have altogether at first?
n = r
(n+65) = 3 (r -23)
n + 65 = 3 (n-23)
n + 65 = 3 n - 69
134 = 2 n
n = r = 67
so
n + r = 134
Let's break down the problem step by step:
1. Let's assume that Nora and Renata had the same number of polished stones at first. Let's call this number "x."
2. After Nora received 65 more stones, she had x + 65 stones.
3. After Renata lost 23 stones, she had x - 23 stones.
4. According to the problem, Nora had three times as many stones as Renata. So we can set up the equation: x + 65 = 3(x - 23).
5. Let's solve this equation for x:
x + 65 = 3x - 69
Subtract x from both sides:
65 = 2x - 69
Add 69 to both sides:
134 = 2x
Divide both sides by 2:
x = 67
6. Therefore, at first Nora and Renata had 67 polished stones each.
7. To find the total number of stones they had altogether at first, we can add their individual numbers:
Total = Nora's stones + Renata's stones
= 67 + 67
= 134
So, Nora and Renata had a total of 134 stones altogether at first.
To solve this problem, we can set up an equation based on the given information. Let's say the initial number of polished stones for both Nora and Renata is represented by 'x'.
After Nora received 65 more stones, she then had 'x + 65' polished stones.
After Renata lost 23 stones, she then had 'x - 23' polished stones.
Given that Nora had three times as many stones as Renata, we can express this as an equation:
x + 65 = 3(x - 23)
Now, let's solve this equation to find the value of 'x'.
Distribute the 3 on the right side of the equation:
x + 65 = 3x - 69
Rearrange the equation by subtracting x from both sides:
65 = 2x - 69
Next, add 69 to both sides of the equation:
65 + 69 = 2x
134 = 2x
Divide both sides of the equation by 2 to isolate 'x':
134/2 = x
67 = x
So the initial number of polished stones for both Nora and Renata was 67.
To find the total number of stones they had altogether at first, we can simply add the initial number of stones for Nora and Renata:
67 + 67 = 134
Therefore, Nora and Renata had altogether 134 polished stones at first.