Question

theos strategy
this is how theo made a hanger with fewer objects.
will the new hanger be balanced?
yes no im not sure

Explanation:

Step1: Analyze original hanger balance

Let the weight of a purple circle be \( p \). On the left side, original weight: \( p + 20 + p + 20 \) (wait, no, looking at the diagram: left side has a purple circle, then a 20, then a purple circle, then a 20 (the striped part). Right side: 20, 20, 20, then 20, 20, 20 (striped). Wait, no, the striped area is removed. So original (before removing striped) left: purple + 20 + purple + 20 (striped). Right: 20 + 20 + 20 + 20 + 20 + 20 (striped). But when we remove the striped part, left becomes purple + 20, right becomes 20 + 20 + 20. Wait, no, maybe better: in the hanger, balance means left weight = right weight. Let's denote the purple circle as \( x \).

Original (including striped): Left: \( x + 20 + x + 20 \) (two purples, two 20s in striped). Right: \( 20 + 20 + 20 + 20 + 20 + 20 \) (six 20s, three in non - striped, three in striped? Wait, no, the diagram: left side (non - striped) has purple, 20; striped has purple, 20. Right side (non - striped) has three 20s; striped has three 20s. So total left: \( x + 20 + x + 20 = 2x + 40 \). Total right: \( 20\times3+20\times3 = 120 \). So \( 2x + 40 = 120 \), so \( 2x = 80 \), \( x = 40 \)? Wait, no, maybe I misread. Wait, the non - striped left: purple (top) and 20 (below it). Striped left: purple and 20. Non - striped right: three 20s (top three). Striped right: three 20s (bottom three). So when we remove the striped part, left becomes purple + 20, right becomes three 20s (20×3 = 60). Now, from original balance (including striped), left (striped + non - striped) = right (striped + non - striped). So (purple + 20 + purple + 20) = (20×3 + 20×3). So \( 2x + 40 = 120 \), so \( 2x = 80 \), \( x = 40 \). Now, after removing striped, left: \( x + 20 = 40 + 20 = 60 \). Right: \( 20×3 = 60 \). So left = right. So the new hanger (after removing striped) has left: purple + 20, right: 20 + 20 + 20. Since \( x = 40 \), 40+20 = 60, and 20×3 = 60. So they are equal. Wait, no, wait: non - striped left: top purple, then 20. So left weight: \( x + 20 \). Non - striped right: three 20s, so 60. And we know from original balance (with striped) that \( 2x + 40 = 120 \) (since striped left is \( x + 20 \), striped right is \( 20×3 = 60 \), so total left \( (x + 20)+(x + 20)=2x + 40 \), total right \( 60 + 60 = 120 \)). So \( 2x=80 \), \( x = 40 \). Then \( x + 20 = 60 \), and right non - striped is 60. So they are equal. So the new hanger is balanced.

Step2: Check the new hanger

After removing the striped objects, left side weight: weight of purple circle + 20. We found purple circle is 40, so 40 + 20 = 60. Right side weight: 20+20+20 = 60. So left = right, so the hanger is balanced.

Answer:

Yes