A flow diagram has a rule that multiplies the input by 7 and then adds 3. If the output is 80, what is the input?

Here are the solutions based on the provided flow diagram and questions:

b. Flow diagram

The rule is "$\times 7 + 3$".

• For input 8: $8 \times 7 + 3 = 56 + 3 = 59$
• For output 80: Let input be $x$. $$x \times 7 + 3 = 80$$ $$7x = 80 - 3$$ $$7x = 77$$ $$x = \frac{77}{7}$$ $$x = 11$$
• For input 6: $6 \times 7 + 3 = 42 + 3 = 45$
• For input 3: $3 \times 7 + 3 = 21 + 3 = 24$
• For input 5: $5 \times 7 + 3 = 35 + 3 = 38$
• For input 1: $1 \times 7 + 3 = 7 + 3 = 10$

The completed flow diagram values are:

• Inputs: $8, 11, 6, 3, 5, 1$
• Outputs: $59, 80, 45, 24, 38, 10$

b.i. What are the input values?
The input values are $\boxed{\text{8, 11, 6, 3, 5, 1}}$.

b.ii. What are the output values?
The output values are $\boxed{\text{59, 80, 45, 24, 38, 10}}$.

b.iii. What is the rule?
The rule is $\boxed{\text{multiply by 7 and add 3 (or } \times 7 + 3)}$.

b.iv. Number sentences:

• $8 \times 7 + 3 = 59$
• $11 \times 7 + 3 = 80$
• $6 \times 7 + 3 = 45$
• $3 \times 7 + 3 = 24$
• $5 \times 7 + 3 = 38$
• $1 \times 7 + 3 = 10$