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Write the rule for fractions with a denominator of 2 to change them to whole numbers and provide a reason. Also, find the rule for the number of sunflowers planted in each row based on the provided pa

Mathematics

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2 parts
3.5

multiply by 2\text{multiply by 2}

4.2

35th row\text{35th row}

Step-by-step solution

7 steps

here are the solutions to the next questions.

Part 3

Part 3.5

multiply by 2\text{multiply by 2}

Write the rule that should be applied to a list of inputs that are fractions with a denominator of 2 to change them to whole numbers. Provide a reason for your answer.

  1. Determine the operation needed to eliminate the denominator.

    To change a fraction with a denominator of 2 into a whole number, you need to multiply it by 2.

  2. Provide a reason.

    Multiplying a fraction by its denominator cancels out the denominator, resulting in the numerator as a whole number. For example, x2×2=x\frac{x}{2} \times 2 = x.

    The rule is to multiply by 2\boxed{\text{multiply by 2}}.
    The reason is that multiplying a fraction by its denominator will cancel out the denominator, leaving the numerator as a whole number.

Part 4

Part 4.1· 3 steps

Complete the table below to show the relationship between the rows and the number of sunflowers planted in each row.

First, let's identify the pattern in the number of sunflowers:
Row 1: 4 sunflowers
Row 2: 7 sunflowers
Row 3: 10 sunflowers
Row 4: 13 sunflowers

  1. Find the common difference between consecutive terms.

    7−4=37 - 4 = 3
    10−7=310 - 7 = 3
    13−10=313 - 10 = 3
    The number of sunflowers increases by 3 for each subsequent row. This is an arithmetic sequence.

  2. Determine the rule for the nn-th row.

    The general formula for an arithmetic sequence is an=a1+(n−1)da_n = a_1 + (n-1)d, where ana_n is the number of sunflowers in row nn, a1a_1 is the number of sunflowers in the first row, and dd is the common difference.
    Here, a1=4a_1 = 4 and d=3d = 3.
    So, an=4+(n−1)3a_n = 4 + (n-1)3
    an=4+3n−3a_n = 4 + 3n - 3
    an=3n+1a_n = 3n + 1

  3. Calculate the number of sunflowers for Row 7.

    For n=7n=7: a7=3(7)+1=21+1=22a_7 = 3(7) + 1 = 21 + 1 = 22.

    The completed table is:

    Row Number12347n
    Number of Sunflowers471013223n+13n+1

Part 4.2

35th row\text{35th row}

Which row will have 106 sunflowers? Show your calculations.

  1. Use the rule an=3n+1a_n = 3n + 1 and set an=106a_n = 106.

    3n+1=1063n + 1 = 106
  2. Solve for nn.

    3n=106−13n = 106 - 1 3n=1053n = 105 n=1053n = \frac{105}{3} n=35n = 35

    The 35th row\boxed{\text{35th row}} will have 106 sunflowers.

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