Question

1. sarah made 42 gingerbread cookies. that is 3 times more cookies than caitlin made. how many cookies did caitlin make? 2. \\( \frac { 1 } { 2 } \\) of sarahs cookies have vanilla icing. \\( \frac { 1 } { 8 } \\) of her cookies have strawberry icing. the rest of her cookies have no icing. what fraction of sarahs cookies have no icing? 3. solve. \\( 27,302 = \\) \\( + 4,651 \\) \\( = 96,207 + 8,699 \\) \\( 54.67 = \\) \\( - 24.1 \\) 4. write the factors for each number. circle the number if it is prime. 12 16 13 17 14 18 15 19 5. the rule is add 0.3 color the path to the gingerbread man by adding 0.3

Explanation:

1.

Step1: Determine the number of cookies Caitlin made

Let \(x\) be the number of cookies Caitlin made. We know that \(42 = 3x\). To find \(x\), we use the formula \(x=\frac{42}{3}\).
$$x = \frac{42}{3}=14$$

2.

Step1: Find the fraction of cookies with no icing

The sum of the fractions of cookies with vanilla and strawberry icing is \(\frac{1}{2}+\frac{1}{8}\). First, find a common denominator (which is \(8\)). Then \(\frac{1}{2}=\frac{4}{8}\), and \(\frac{4}{8}+\frac{1}{8}=\frac{4 + 1}{8}=\frac{5}{8}\).
Since the total fraction of all cookies is \(1\) (or \(\frac{8}{8}\)), the fraction of cookies with no icing is \(1-\frac{5}{8}\). Using the formula \(1-\frac{5}{8}=\frac{8-5}{8}\)
$$1-\frac{5}{8}=\frac{3}{8}$$

3.

Step1: Solve \(27302=\square + 4651\)

Use the formula \(\square=27302 - 4651\).
$$27302-4651 = 22651$$

Step2: Solve \(\square=96207+8699\)

Use the formula \(\square=96207 + 8699\)
$$96207+8699=104906$$

Step3: Solve \(54.67=\square - 24.1\)

Use the formula \(\square=54.67+24.1\)
$$54.67 + 24.1=78.77$$

4.

Step1: Find factors of \(12\)

Use the formula \(n = ab\). For \(n = 12\), \(1\times12=12\), \(2\times6 = 12\), \(3\times4=12\). So factors of \(12\) are \(1,2,3,4,6,12\) (not prime).

Step2: Find factors of \(16\)

For \(n = 16\), \(1\times16=16\), \(2\times8 = 16\), \(4\times4=16\). So factors of \(16\) are \(1,2,4,8,16\) (not prime).

Step3: Find factors of \(13\)

For \(n = 13\), \(1\times13=13\). So factors of \(13\) are \(1,13\) (prime).

Step4: Find factors of \(17\)

For \(n = 17\), \(1\times17=17\). So factors of \(17\) are \(1,17\) (prime).

Step5: Find factors of \(14\)

For \(n = 14\), \(1\times14=14\), \(2\times7=14\). So factors of \(14\) are \(1,2,7,14\) (not prime).

Step6: Find factors of \(18\)

For \(n = 18\), \(1\times18=18\), \(2\times9=18\), \(3\times6=18\). So factors of \(18\) are \(1,2,3,6,9,18\) (not prime).

Step7: Find factors of \(15\)

For \(n = 15\), \(1\times15=15\), \(3\times5=15\). So factors of \(15\) are \(1,3,5,15\) (not prime).

Step8: Find factors of \(19\)

For \(n = 19\), \(1\times19=19\). So factors of \(19\) are \(1,19\) (prime).

5.

Step1: Follow the rule \(y=x + 0.3\)

Start with \(x = 3.1\), then \(y_1=3.1+0.3 = 3.4\), \(y_2=3.4+0.3=3.7\), \(y_3=3.7+0.3 = 4\) (not in the path), \(y_4 = 4.3\) (not following the rule from \(3.7\)), \(y_5=4.6\) (not following the rule from \(3.7\)), \(y_6=4.9\) (not following the rule from \(3.7\)), \(y_7=5.2\) (not following the rule from \(3.7\)), \(y_8=5.5\) (not following the rule from \(3.7\)), \(y_9=5.8\) (not following the rule from \(3.7\)), \(y_{10}=6.1\) (not following the rule from \(3.7\)). Another path: Start with \(3.1\), then \(3.1+0.3=3.4\), \(3.4+0.3 = 3.7\), \(3.7+0.3=4\) (wrong), start again. Start with \(3.1\), then \(3.1+0.3 = 3.4\), \(3.4+0.3=3.7\), \(3.7+0.3 = 4\) (wrong). Correct path: \(3.1\to4.4\) (wrong, \(3.1+0.3
eq4.4\)), \(3.1\to4.7\) (wrong). The correct sequence (by adding \(0.3\) each time): \(3.1\to3.4\to3.7\) (but no correct continuation in the grid as per strict addition, assume intended path: \(3.1\) (start), then \(3.1+0.3 = 3.4\), \(3.4+0.3=3.7\), \(3.7+0.3 = 4\) (not, but if we consider \(3.1\to4.4\) (wrong), actual correct numerical - based on addition: \(3.1+0.3n\). \(3.1+0.3\times10=6.1\) (last number). The path is \(3.1\to3.4\to3.7\to4\) (not, but in terms of grid - assume coloring \(3.1\), then adding \(0.3\) each step: \(3.1\), \(3.4\), \(3.7\), \(4\) (no), \(4.3\) (no), \(4.6\) (no), \(4.9\) (no), \(5.2\) (no), \(5.5\) (no), \(5.8\) (no), \(6.1\)…

Answer:

1. \(14\) cookies
2. \(\frac{3}{8}\)
3. \(22651\), \(104906\), \(78.77\)

4.

• \(12\): \(1,2,3,4,6,12\)
• \(16\): \(1,2,4,8,16\)
• \(13\): \(1,13\) (circle \(13\))
• \(17\): \(1,17\) (circle \(17\))
• \(14\): \(1,2,7,14\)
• \(18\): \(1,2,3,6,9,18\)
• \(15\): \(1,3,5,15\)
• \(19\): \(1,19\) (circle \(19\))

1. Color the hexagons with \(3.1\) (start) and \(6.1\) (end) (assuming the intended path based on the rule \(y=x + 0.3\) with \(x = 3.1\) and \(y=6.1\) (\(n = 10\) times addition of \(0.3\))