Here are the solutions to the problems on the quiz sheet.
1. Write each decimal below as a fraction in simplest form.
a. -2.498
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Step 1: Write the decimal as a fraction over 1000 because there are three decimal places.
$$-2.498 = -\frac{2498}{1000}$$
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Step 2: Simplify the fraction. Both numbers are even, so divide by 2.
$$2498 \div 2 = 1249$$
$$1000 \div 2 = 500$$
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Result: $-\frac{1249}{500}$ (This cannot be simplified further).
b. 0.12
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Step 1: Write the decimal as a fraction over 100.
$$0.12 = \frac{12}{100}$$
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Step 2: Simplify. Both numbers are divisible by 4.
$$12 \div 4 = 3$$
$$100 \div 4 = 25$$
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Result: $\frac{3}{25}$
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2. Write each fraction as a decimal.
a. $\frac{17}{2}$
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Step 1: Divide 17 by 2.
$$17 \div 2 = 8.5$$
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Result: $8.5$
b. $\frac{18}{100}$
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Step 1: Since the denominator is 100, move the decimal point in the numerator two places to the left.
$$18 \rightarrow 0.18$$
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Result: $0.18$
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3. Determine if each number is rational or irrational.
Rule: A number is
Rational if it can be written as a fraction (integers, terminating decimals, repeating decimals). It is
Irrational if it cannot (non-terminating, non-repeating decimals, square roots of non-perfect squares).
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$\frac{2\pi}{3}$: Contains $\pi$ (pi), which is irrational. Multiplying/dividing an irrational number by a rational number keeps it irrational. ->
Irrational
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-41.37: This is a terminating decimal. It can be written as a fraction ($-\frac{4137}{100}$). ->
Rational
*
$\sqrt{81}$: The square root of 81 is 9 ($9 \times 9 = 81$). 9 is an integer. ->
Rational
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-2.8: This is a terminating decimal. ->
Rational
*
$\sqrt{80}$: 80 is not a perfect square (it falls between $64$ and $81$). Its decimal goes on forever without repeating. ->
Irrational
*
$\frac{2\pi}{5}$: Contains $\pi$. ->
Irrational
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4. Circle all of the rational numbers.
Let's check each option:
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a. $4\pi$: Irrational (contains $\pi$).
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b. $\frac{5}{7}$: Rational (it is a fraction of integers).
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c. $\sqrt{25}\pi$: $\sqrt{25} = 5$, so this is $5\pi$. Irrational (contains $\pi$).
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d. -2.32875321...: The "..." usually indicates the pattern continues. In these quizzes, long random-looking decimals with ellipses are typically
Irrational. If it were rational, it would either stop or show a repeating bar.
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e. $7.\overline{2}$: The bar over the 2 means it repeats ($7.2222...$). Repeating decimals are
Rational.
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f. $\sqrt{121}$: $\sqrt{121} = 11$. Integers are
Rational.
The rational numbers are: b, e, and f.
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Final Answer:
1. Decimals to Fractions:
a. $-\frac{1249}{500}$
b. $\frac{3}{25}$
2. Fractions to Decimals:
a. $8.5$
b. $0.18$
3. Rational vs. Irrational Table:
* $\frac{2\pi}{3}$:
Irrational
* $-41.37$:
Rational
* $\sqrt{81}$:
Rational
* $-2.8$:
Rational
* $\sqrt{80}$:
Irrational
* $\frac{2\pi}{5}$:
Irrational
4. Circle the Rational Numbers:
*
b. $\frac{5}{7}$
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e. $7.\overline{2}$
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f. $\sqrt{121}$
Parent Tip: Review the logic above to help your child master the concept of identify rational numbers worksheet.
