Question

week 18
monday
evaluate and simplify.
$2\frac{1}{6} - \frac{1}{2} =$
for every 4 dogs, there are 5 cats in a shelter. if there are 2 more dogs than cats, how many total animals are there?
tuesday
determine the value.
the opposite of -3.
the absolute value of 9.
the opposite of the opposite of 10.
evaluate.
$3 \div \frac{1}{2} =$

Explanation:

Step1: Convert mixed number to improper fraction

$2\frac{1}{6} = \frac{2\times6 + 1}{6} = \frac{13}{6}$

Step2: Find common denominator for subtraction

$\frac{1}{2} = \frac{3}{6}$

Step3: Subtract the fractions

$\frac{13}{6} - \frac{3}{6} = \frac{10}{6}$

Step4: Simplify the result

$\frac{10}{6} = 1\frac{2}{3}$

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Step1: Find opposite of -3

Opposite of $-3$ is $3$

Step2: Find absolute value of 9

$|9| = 9$

Step3: Find opposite of opposite of 10

Opposite of $10$ is $-10$, opposite of $-10$ is $10$

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Step1: Define variables for ratio

Let number of dogs = $4x$, number of cats = $5x$

Step2: Set up equation for animal count

$4x = 5x + 2$

Step3: Solve for $x$

$4x - 5x = 2 \implies -x = 2 \implies x = -2$

Step4: Calculate total animals

Number of dogs: $4\times(-2) = -8$ (Note: This indicates a typo in the problem; assuming it is 2 more cats than dogs, use $5x = 4x + 2$:
$5x - 4x = 2 \implies x=2$
Dogs: $4\times2=8$, Cats: $5\times2=10$
Total animals: $8+10=18$

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Step1: Rewrite division as multiplication by reciprocal

$3 \div \frac{1}{2} = 3 \times 2$

Step2: Calculate the product

$3 \times 2 = 6$

Answer:

1. $2\frac{1}{6} - \frac{1}{2} = 1\frac{2}{3}$
2. Opposite of -3: 3; Absolute value of 9: 9; Opposite of opposite of 10: 10
3. Total animals: 18 (corrected for logical animal counts)
4. $3 \div \frac{1}{2} = 6$