Decimals with Fractions - SSAT Upper Level Quantitative

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Question

Which of the following fractions is closest to 0.6?

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Answer

For each fraction, divide each numerator by its denominator to get its decimal representation, then subtract to find its difference from 0.6.

$$\frac{4}{7}$ \approx 0.5714 ; 0.6 - 0.5714 =0.0286$

$$\frac{5}{8}$=0.625; 0.625-0.6 = 0.025$

$$\frac{9}{16}$ = 0.5625; 0.6-0.5625=0.0375$

$$\frac{2}{3}$\approx 0.6667; 0.6667-0.6=0.0667$

$$\frac{13}{20}$ = 0.65; 0.65-0.6=0.05$

$\frac{5}{8}$ is the closest to 0.6 of the five.

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Question

Express $\frac{8}{15}$ as a decimal.

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Answer

Divide 8 by 15:

$$\frac{8}{15}$ = 8 \div 15 = 0.5333...$

The 3 repeats, so the correct choice is $0.5$\overline{3}$$ .

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Question

Give the decimal equivalent of $\frac{11}{18}$ .

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Answer

Divide 11 by 18:

$$\frac{11}{18}$ = 11 \div 18 = 0.61111...$

The 1 repeats infinitely, so this can be rewritten as $0.6$\overline{1}$$ .

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Question

Give the decimal equivalent of $\frac{5}{27}$ .

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Answer

Divide 5 by 27:

$$\frac{5}{27}$ = 5 \div 27 = 0.185185185...$

The group "185" repeats infinitely, so this can be written as $0.$\overline{185}$$ .

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Question

Give the decimal equivalent of $\frac{7}{36}$ .

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Answer

Divide 7 by 36:

$$\frac{7}{36}$ = 7 \div 36 = 0.19444444...$

The 4 repeats infinitely, so this can be rewritten as $0.19$\overline{4}$$ .

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Question

Write the value of this expression as a decimal:

$\frac{1}{3}$ + $\frac{1}{4}$ + $\frac{1}{2}$

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Answer

Simplify the sum by taking the least common denominator, which is 12:

$\frac{1}{3}$ + $\frac{1}{4}$ + $\frac{1}{2}$

$= $\frac{1\cdot 4}{3\cdot 4}$ + $\frac{1\cdot 3}{4\cdot 3}$ + $\frac{1\cdot 6}{2\cdot 6}$$

$= $\frac{ 4}{12}$ + $\frac{3}{12}$ + $\frac{6}{12}$ = $\frac{13}{12}$$

Divide 13 by 12:

$\frac{13}{12}$ = 1. 08333333... = 1.08$\overline{3}$

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Question

Let $a = $\frac{3}{5}$$ , $b= $\frac{2}{3}$$ , $c = $\frac{4}{7}$$

Which of the following is closest to ?

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Answer

$= $\frac{3}{5}$+ $\frac{2}{3}$+ $\frac{4}{7}$$

$= $\frac{3 \times 21}{5 \times 21}$+ $\frac{2\times 35}{3 \times 35}$+ $\frac{4\times 15}{7 \times 15}$$

$= $\frac{63}{105}$+ $\frac{70}{105}$+ $\frac{60}{105}$ = $\frac{193}{105}$$

$= 193 \div 105 \approx 1.84$

The correct choice is 1.8.

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Question

Simplify and express as a decimal:

$$\frac{4\frac{2}{3}$}{5$\frac{3}{5}$}$

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Answer

$\frac{4\frac{2}{3}$}{5$\frac{3}{5}$} = 4$\frac{2}{3}$ \div 5$\frac{3}{5}$ = $\frac{14}{3}$ \div $\frac{28}{5}$ = $\frac{14}{3}$ \times $\frac{5}$ {28} = $\frac{1}{3}$ \times $\frac{5}$ {2} = $\frac{5}{6}$

$$\frac{5}{6}$= 5 \div 6 = 0.8333...$ ,

so the correct response is $0.8$\overline{3}$$ .

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Question

Let $a =3 $\frac{3}{5}$$ and $b=1 $\frac{5}{6}$$

Which of the following is closest to ?

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Answer

$a - b = 3$\frac{3}{5}$ - 1$\frac{5}{6}$$

$= $\frac{18}{5}$ - $\frac{11}{6}$$

$= $\frac{18\times 6 }{5 \times 6}$ - $\frac{11\times 5}{6\times 5}$$

$= $\frac{108}{30}$ - $\frac{55}{30}$$

$= $\frac{53}{30}$$

$= 53 \div 30 \approx 1.77$

The correct choice is 1.8.

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Question

Let $a =1 $\frac{4}{9}$$ , $b = 1$\frac{3}{8}$$ , and $c= $\frac{1}{2}$$ .

Which of the following is closest to ?

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Answer

$a bc=1 $\frac{4}{9}$ \times 1$\frac{3}{8}$ \times $\frac{1}{2}$= $\frac{13}{9}$ \times $\frac{11}{8}$ \times $\frac{1}{2}$= $\frac{143}{144}$= 143 \div 144 \approx 0.99$

The correct response is 1.

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Question

Define an operation $\blacklozenge$ as follows:

For all real numbers :

$a\blacklozenge b = $\frac{ab}{a+b}$$

Which of the following is equal to $\frac{1}{2}$ \blacklozenge $\frac{1}{3}$ ?

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Answer

$a\blacklozenge b = $\frac{ab}{a+b}$$

$\frac{1}{2}$ \blacklozenge $\frac{1}{3}$=\frac{\frac{1}{2}$ \times $\frac{1}{3}$}{$\frac{1}{2}$ +\frac{1}{3}$}=\frac{\frac{1}{6}$ }{$\frac{5}{6}$ } = $\frac{1}{6}$ \div $\frac{5}{6}$ = $\frac{1}{6}$ \times $\frac{6}$ {5} = $\frac{1}{1}$ \times $\frac{1}$ {5} = $\frac{1}{5}$

$$\frac{1}{5}$ = 1 \div 5 = 0.2$ ,

is the correct response.

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Question

Define an operation $\blacklozenge$ as follows:

For all real numbers :

$a\blacklozenge b = $a^{2}$ - $b^{2}$$

Which of the following is closest to $\frac{1}{2}$ \blacklozenge $\frac{1}{3}$ ?

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Answer

$a\blacklozenge b = $a^{2}$ - $b^{2}$$

$\frac{1}{2}$ \blacklozenge $\frac{1}{3}$ =\left ( $\frac{1}{2}$ \right ) ^{2} - \left ( $\frac{1}{3}$ \right $)^{2}$= $\frac{1}{4}$ - $\frac{1}{9}$= $\frac{9}{36}$ - $\frac{4}{36}$= $\frac{5}{36}$

$= 5 \div 36 = 0.13888...$

The correct choice is 0.15.

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Question

Which of the following is a true statement?

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Answer

Divide 7 by 13. The result is as follows:

The correct choice is that $0.53 < $\frac{7}{13}$ < 0.54$ .

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Question

Which of the following is closest to the sum of two thirds and four fifths?

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Answer

$\frac{2}{3}$ + $\frac{4}{5}$ = $\frac{2 \cdot 5}{3\cdot 5}$ + $\frac{3\cdot 4}{3\cdot 5}$ = $\frac{10}{15}$ + $\frac{12}{15}$ =\frac{22}{15}$

Divide 22 by 15:

Of the five choices, the closest is 1.45.

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Question

Which of the following is a true statement?

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Answer

Divide 19 by 27:

$0.70 < $\frac{19}{27}$< 0.71$ ,

so

$-0.71< -$\frac{19}{27}$< -0.70$ .

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Question

Which of the following falls between 0.32 and 0.33?

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Answer

Since all of the fractions have 77 as a denominator, one way to solve this problem is to let be the numerator of the correct fraction. Therefore, the equivalent problem is to find integer such that

$0.32 < $\frac{N}{77}$ < 0.33$

Multiply each expression by :

$0.32 \cdot 77 < $\frac{N}{77}$ \cdot 77 < 0.33 \cdot 77$

Therefore, , making $\frac{25}{77}$ the correct choice.

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Question

Rewrite as a fraction with whole numerator and denominator in lowest terms:

$\frac{1.28}{0.4}$

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Answer

Simplify as follows:

$\frac{1.28}{0.4}$ = $\frac{1.28 \times 100}{0.4 \times 100}$ = $\frac{128}{40}$ = $\frac{128 \div 8}{40 \div 8}$ = $\frac{16}{5}$

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Question

Rewrite as a fraction with whole numerator and denominator in lowest terms:

$\frac{3.5}{2.1}$

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Answer

$\frac{3.5}{2.1}$ = $\frac{3.5 \times 10}{2.1 \times 10}$ = $\frac{35}{21}$ = $\frac{35 \div 7}{21\div 7}$ = $\frac{5}{3}$

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Question

Simplify:

$0.35+\frac{0.75}{0.2}$-0.4*$\frac{0.5}{0.1}$$

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Answer

Begin by multiplying all of your decimal fractions by $\frac{100}{100}$ :

$0.35+\frac{0.75}{0.2}$$\frac{100}{100}$-0.4$\frac{0.5}{0.1}$$\frac{100}{100}$$

Simplify:

$0.35+\frac{75}{20}$-0.4$\frac{50}{10}$=0.35+\frac{15}{4}$-0.45$

Now perform the multiplication:

$0.35+\frac{15}{4}$-2$

The easiest thing to do next is to subtract from :

$0.35+\frac{15}{4}$-2=\frac{15}{4}$-1.65$

Next, convert into the fraction $\frac{165}{100}$ :

$\frac{15}{4}$-$\frac{165}{100}$

Now, the common denominator can be :

$\frac{15}{4}$$\frac{25}{25}$-$\frac{165}{100}$=\frac{375}{100}$-$\frac{165}{100}$

Simplify:

$\frac{375}{100}$-$\frac{165}{100}$=\frac{210}{100}$=\frac{21}{10}$

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Question

Rewrite the following fraction in simplest form: $\frac{0.42}{0.84}$

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Answer

In order to rewrite $\frac{0.42}{0.84}$ in simplest form, multiply by a form of that makes the fraction easier to reduce - in this case , $\frac{100}{100}$ :

$\frac{0.42\times 100}{0.84\times 100}$=\frac{42}{84}$=\frac{1}{2}$

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