How to multiply fractions - ISEE Middle Level Quantitative Reasoning

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Question

$\small \frac{3}{15}\cdot \frac{4}{5}$

Answer

When multiplying fractions you simply multiply the numerators to find the numerator of the product, and multiply the denominators to find the denominator of the product.

$\small 3\cdot 4=12$

$\small 5\cdot 15=75$

So, our product is $\small \frac{12}{75}$ .

Since both of these numbers are divisible by three, we simplify.

$\small \frac{12}{3}=4$

$\small \frac{75}{3}= 25$

Our final answer is $\small \frac{4}{25}$ .

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Question

Which is the greater quantity?

(a) $3 \frac{1}{3} \times 4 \frac{1}{5}$

(b) $12 \frac{1}{15}$

Answer

Rewrite the mixed numbers as improper fractions, then multiply across:

$3 \frac{1}{3} = \frac{3\times 3 + 1}{3} = \frac{10}{3}$

$4 \frac{1}{5} = \frac{4 \times 5 + 1}{5} = \frac{21}{5}$

$3 \frac{1}{3} \times 4 \frac{1}{5} = \frac{10}{3} \times \frac{21}{5} = \frac{2}{1} \times \frac{7}{1} = 14$

$14 > 12 \frac{1}{5}$

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Question

Which is the greater quantity?

(a) $6 \frac{5}{6} \times 6$

(b)

Answer

Rewrite the factors as improper fractions, then multiply across:

$6 \frac{5}{6} = \frac{6\times 6 + 5}{6} = \frac{36 + 5}{6} = \frac{41}{6}$

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

$6 \frac{5}{6} \times 6 =\frac{41}{6}\times \frac{ 6}{1}=\frac{41}{1}\times \frac{ 1}{1} = 41$

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Question

$\frac{7}{4} k = \frac{7}{10}$

Which is the greater quantity?

(a)

(b)

Answer

(a)

$0.75 t \div 0.75 = 0.3 \div 0.75$

$t= 0.3 \div 0.75$

Divide by moving the decimal point right two places in both numbers:

$t= 30 \div 75 = 0.4$

(b) $\frac{7}{4} k = \frac{7}{10}$

$\frac{4} {7} \cdot \frac{7}{4} k =\frac{4} {7} \cdot \frac{7}{10}$

$k =\frac{4} {7} \cdot \frac{7}{10}$

Cross-cancel:

$k =\frac{2} {1} \cdot \frac{1}{5} = \frac{2} {5}$

$0.4 = \frac{4}{10 } = \frac{4\div 2}{10\div 2 } = \frac{2}{5}$ , so

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Question

$a=1.5,\ b = 4.5$

Which is the greater quantity?

(A)

(B)

Answer

Solve for (A):

$ab = 1.5 \cdot 4.5 = 6.75$

Solve for (B):

Since , and (A) is greater.

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Question

Which is the greater quantity?

(A) $\left ( \frac{3}{5} \right )^{3}$

(B) $0.6^{2}$

Answer

$\frac{3}{5} = 3 \div 5 = 0.6$ , so

$\left ( \frac{3}{5} \right )^{3} = 0.6^{3} = 0.6 \times 0.6 \times 0.6 = 0.36 \times 0.6 = 0.216$

$0.6 ^{2} = 0.6 \times 0.6 = 0.36$

, so $0.6^{2} > \left ( \frac{3}{5} \right )^{3}$ , making (B) greater.

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Question

Which is the greater quantity?

(A)

(B) $\left ( \frac{4}{5} \right )^{3}$

Answer

$\left ( \frac{4}{5} \right )^{3} = 0.8 ^{3} = 0.8 \times 0.8 \times 0,8 = 0.64\times 0,8 = 0.512$

Since , $\left ( \frac{4}{5} \right )^{3} > 0.5$ , so (B) is greater.

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Question

Which is the greater quantity?

(A) $\left ( \frac{7}{25} \right )^{2}+ \left ( \frac{24}{25} \right )^{2}$

(B) $0.8^{2} + 0.6^{2}$

Answer

$\left ( \frac{7}{25} \right )^{2}+ \left ( \frac{24}{25} \right )^{2}$

$=\frac{7^{2}}{25^{2}} + \frac{24^{2}}{25^{2}}$

$=\frac{49}{625 } + \frac{576}{625 } = \frac{49+576}{625 } = \frac{625}{625 } = 1$

$0.8 ^{2} + 0.6^{2}$

$=0.8 \times 0.8 + 0.6 \times 0.6$

The expressions are equal.

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Question

Which is the greater quantity?

(A) $\left (\frac{3}{5} \right )^{2}$

(B) $1 - \left ( \frac{2}{5} \right )^{2}$

Answer

$\left (\frac{3}{5} \right )^{2} = \frac{3^{2}}{5^{2}} = \frac{9}{25}$

$1 - \left ( \frac{2}{5} \right )^{2} = 1- \frac{2^{2}}{5^{2}} = 1- \frac{4}{25} = \frac{25}{25}- \frac{4}{25} = \frac{25-4}{25}= \frac{21}{25}$

$\frac{21}{25} > \frac{9}{25}$ , so (B) is greater

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Question

$\tfrac{3}{4} \times 12 =$

Answer

When dealing with problems that involve fractions and whole numbers, first convert your whole number into a fraction. You can easily do this by putting the whole number over 1.

$12 \rightarrow\tfrac{12}{1}$

$\tfrac{3}{4}\times12=\tfrac{3}{4}\times\tfrac{12}{1}$

When multiplying, multiply the numerators by each other and the denominators by each other.

$\tfrac{3}{4} \times\tfrac{12}{1} = \tfrac{3\times12}{4\times1}=\tfrac{36}{4}$

Always reduce a fraction when possible. In this case, the numerator and denominator of $\tfrac{36}{4}$ are both divisible by 4. So, it reduces to 9.

$\tfrac{36}{4}=\tfrac{36\div4}{4\div4}=\tfrac{9}{1}=9$

$\tfrac{36}{4} \rightarrow\tfrac{9}{1}\rightarrow9$

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Question

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

Answer

When multiplying fractions, simply multiply the numerators by each other and the denominators by each other.

$\frac{1}{2} \times \frac{2}{3} = \frac{1\times2}{2\times3}=\frac{2}{6}$

Always reduce fractions when possible. In this case, both the numerator and denominator are divisible by 2.

$\frac{2}{6}=\frac{2\div2}{6\div2}=\frac{1}{3}$

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Question

$\frac{4}{7} \times \frac{1}{2} =$

Answer

When multiplying fractions, multiply the numerators and denominators.

$\frac{4}{7} \times \frac{1}{2} =\frac{4\times1}{7\times2}=\frac{4}{14}$

When possible, always reduce your fraction. In this case, both 4 and 14 are divisible by 2.

$\frac{4}{14}\rightarrow\frac{2}{7}$

The result is your answer.

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Question

$3 \times \tfrac{1}{3} =$

Answer

When dealing with fractions and whole numbers, first convert the whole number into a fraction. This is easily done by putting the whole number over 1.

$3\rightarrow\tfrac{3}{1}$

When multiplying fractions, multiply both the numerators and denominators.

$\frac{3}{1}\times\frac{1}{3} =\frac{3\times1}{1\times3}=\frac{3}{3}$

When possible, always reduce your fraction. In this case, since the numerator and denominator is the same, the result is 1.

$\tfrac{3}{3}\rightarrow1$

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Question

The above pentagon has area 100. Which is the greater quantity?

(a) The area of the white region

(b) 25

Answer

Of the five triangles of equal area, one of them is white - this is $\frac{1}{5 }$ of the pentagon. The pentagon has area 100, so the area of the white triangle is

$\frac{1}{5 } \cdot 100 = \frac{100}{5 } = 20$ ,

which is less than 25.

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Question

Note: figure NOT drawn to scale

Above are a rectangle and a triangle, each of which is divided equally into smaller shapes. Which is the greater quantity?

(a) The area of the shaded portion of the rectangle

(b) The area of the shaded portion of the triangle

Answer

It cannot be determined which of the figures has the greater shaded area, because nothing is given about the areas, or even the comparative areas, of the rectangle and the triangle.

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