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A flow diagram shows a rule that transforms input values into output values. Some input and output values are missing. Determine the input values, the output values, and the rule.

Mathematics

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Answers

4 parts
i

5, 2, 6, 9, 7, 11\text{5, 2, 6, 9, 7, 11}

ii

21, 6, 26, 41, 31, 51\text{21, 6, 26, 41, 31, 51}

iv

Multiply by 5, then subtract 4 (or x→5x−4)\text{Multiply by 5, then subtract 4 (or } x \to 5x - 4)

vi

-100, -40, -120, -180, -140, -220\text{-100, -40, -120, -180, -140, -220}

Step-by-step solution

6 steps

Here are the solutions based on the flow diagram:

Part i· 2 steps

What are the input values?

  1. Identify the given input values from the diagram.

    The given input values are $5, 2, 7, 11$.

  2. Use the rule to find the missing input values.

    The rule is "multiply by 5, then subtract 4". So, if xx is the input and yy is the output, the rule is y=5x−4y = 5x - 4.
    We have output values 26 and 41 with missing inputs.
    For output 26:

    5x−4=265x - 4 = 26 5x=26+45x = 26 + 4 5x=305x = 30 x=305x = \frac{30}{5} x=6x = 6

    For output 41:

    5x−4=415x - 4 = 41 5x=41+45x = 41 + 4 5x=455x = 45 x=455x = \frac{45}{5} x=9x = 9

    The complete list of input values is 5,2,6,9,7,115, 2, 6, 9, 7, 11.

    The input values are: 5, 2, 6, 9, 7, 11\boxed{\text{5, 2, 6, 9, 7, 11}}

Part ii

21, 6, 26, 41, 31, 51\text{21, 6, 26, 41, 31, 51}

What are the output values?

  1. Identify the given output values from the diagram.

    The given output values are $26, 41$.

  2. Use the rule to find the missing output values.

    The rule is y=5x−4y = 5x - 4.
    For input 5:

    y=5(5)−4=25−4=21y = 5(5) - 4 = 25 - 4 = 21

    For input 2:

    y=5(2)−4=10−4=6y = 5(2) - 4 = 10 - 4 = 6

    For input 7:

    y=5(7)−4=35−4=31y = 5(7) - 4 = 35 - 4 = 31

    For input 11:

    y=5(11)−4=55−4=51y = 5(11) - 4 = 55 - 4 = 51

    The complete list of output values is 21,6,26,41,31,5121, 6, 26, 41, 31, 51.

    The output values are: 21, 6, 26, 41, 31, 51\boxed{\text{21, 6, 26, 41, 31, 51}}

Part iv

Multiply by 5, then subtract 4 (or x→5x−4)\text{Multiply by 5, then subtract 4 (or } x \to 5x - 4)

What is the rule?

The rule shown in the flow diagram is to multiply the input by 5, then subtract 4.

The rule is: Multiply by 5, then subtract 4 (or x→5x−4)\boxed{\text{Multiply by 5, then subtract 4 (or } x \to 5x - 4)}

Part v

Number sentences

Based on the input and output values and the rule y=5x−4y = 5x - 4:

5×5−4=215 \times 5 - 4 = 21 5×2−4=65 \times 2 - 4 = 6 5×6−4=265 \times 6 - 4 = 26 5×9−4=415 \times 9 - 4 = 41 5×7−4=315 \times 7 - 4 = 31 5×11−4=515 \times 11 - 4 = 51

Part vi

-100, -40, -120, -180, -140, -220\text{-100, -40, -120, -180, -140, -220}

What will the output values be if the rule is −4×5-4 \times 5?

  1. Interpret the new rule.

    The rule is given as −4×5-4 \times 5. This simplifies to −20-20. So, the new rule is to multiply the input by −20-20.
    Let xx be the input, the new rule is y=−20xy = -20x.

  2. Use the input values from the original diagram ($5, 2, 6, 9, 7, 11$) with the new rule.

    For input 5:

    y=−20×5=−100y = -20 \times 5 = -100

    For input 2:

    y=−20×2=−40y = -20 \times 2 = -40

    For input 6:

    y=−20×6=−120y = -20 \times 6 = -120

    For input 9:

    y=−20×9=−180y = -20 \times 9 = -180

    For input 7:

    y=−20×7=−140y = -20 \times 7 = -140

    For input 11:

    y=−20×11=−220y = -20 \times 11 = -220

    The output values will be: -100, -40, -120, -180, -140, -220\boxed{\text{-100, -40, -120, -180, -140, -220}}

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