Question

5
based on the hanger diagram, what is the value
of the heart? (1 point)

= 6 \t= \t=

6

10

11

14

Explanation:

Step1: Calculate heart's value

The heart balances two 6s, so:
$\text{Heart} = 6 + 6 = 12$
Wait, correct: no, total hanger is 44. Let's redo:

Step1: Find heart's value first

Heart equals two 6s:
$\text{Heart} = 6 + 6 = 12$
Wait no, total left: 2 circles, right: heart. Total 44.

Step1: Compute heart's value

Heart balances two 6s:
$\text{Heart} = 6 + 6 = 12$
Wait no, total weight: 2 circles + heart = 44.

Step1: Calculate heart's value

Heart = $6 + 6 = 12$

Step2: Set up equation for circles

Let $c$ = circle value. $2c + 12 = 44$

Step3: Solve for circle value

$2c = 44 - 12 = 32$
$c = \frac{32}{2} = 16$
Wait, no, question asks for heart. Wait, heart is 12? No, options have 14,10,6,11. Wait, I misread: the hanger top is 44, left is two circles, right is heart, and heart's hanger has two 6s. Wait, no: the right side is heart, which supports two 6s. So heart's value equals the sum of the two 6s? No, no: the hanger is balanced, so left side (two circles) equals right side (heart). And the heart's own hanger: heart equals two 6s? No, no, the total top hanger is 44, so total weight is 44, meaning left + right = 44? No, balanced hanger means left weight = right weight, and total is 44, so left = right = 22. Oh! That's the mistake.

Step1: Split total weight equally

Balanced hanger: left = right = $\frac{44}{2} = 22$

Step2: Calculate heart's value

Heart balances two 6s: $\text{Heart} = 6 + 6 = 12$? No, no: right side is heart, which is 22? No, no, the diagram: top hanger has left: two circles, right: heart. The heart has a sub-hanger with two 6s, so heart's value equals the sum of the two 6s? No, no, the sub-hanger means heart's weight equals the two 6s. Wait, no, the top hanger's total is 44, so left (two circles) + right (heart) = 44? No, balanced hanger means left weight = right weight, so each side is 22. So right side is heart = 22? But heart has a sub-hanger with two 6s, so heart = 6+6=12? That contradicts. Wait, no, the top number 44 is the total weight of both sides. So left (2 circles) + right (heart) = 44. And heart = 6 + 6 = 12. Then 2 circles = 44 -12=32, circle=16. But the question asks for heart, which is 12, but 12 is not an option. Wait, I misread the sub-hanger: the sub-hanger has two 6s, so heart's value is equal to one 6? No, no, the sub-hanger is balanced, so heart = 6? No, options have 6. Wait no, the diagram: the right side is heart, which is holding a hanger with two 6s, so heart's weight is equal to the sum of the two 6s. So heart = 6+6=12. But 12 is not an option. Wait, maybe the top 44 is the weight of each side? So left side (two circles) =44, right side (heart)=44? No, that can't be. Wait, the options are 6,10,11,14. Let's try: if heart is 14, then two circles = 44-14=30, circle=15. No. If heart is 10, two circles=34, circle=17. No. If heart is 11, two circles=33, circle=16.5. No. If heart is 6, two circles=38, circle=19. No. Wait, maybe the top hanger: left is one circle, right is heart? No, diagram shows two circles. Wait, no, maybe the total weight is 44, so left (circle) + right (heart) =44, and heart=6+6=12, circle=32. No. Wait, maybe the sub-hanger: heart is equal to one 6, so heart=6, then two circles=44-6=38, circle=19. But why is the sub-hanger two 6s? Oh! Wait, maybe the heart's weight is equal to the two 6s, so heart=6+6=12, but the top hanger is left (two circles) = right (heart), so two circles=12, circle=6, total weight=24, but top says 44. No. Wait, I must have misread the top number. Oh! Wait, the top number is 44, so total weight is 44, meaning left side + right side =44, and left side = right side, so each is 22. So right side is heart=22, but heart has sub-hanger with two 6s, so 22=6+6? No, that's not. Wait, maybe the sub-hanger means heart +6+6=44? No, left side is two circles, right side is heart +6+6. So 2c = heart +12, and 2c + heart +12=44. Then 2*(heart+12) + heart +12=44? No, 2c = heart +12, so total is (heart+12) + (heart+12)=44, so 2*(heart+12)=44, heart+12=22, heart=10. Oh! That's the correct reading. I misread the diagram: the right side is heart holding two 6s, so right side total weight is heart +6+6, left side is two circles, and total hanger is balanced, so left = right, and total weight is 44, so left + right=44, so left=right=22. So right side: heart +6+6=22.

Step1: Set right side weight to 22

Balanced hanger: right side = $\frac{44}{2}=22$

Step2: Solve for heart value

Heart + 6 + 6 = 22
$\text{Heart} = 22 - 12 = 10$

Answer:

10 (Option: 10)