Questions & Answers: "Fractions and Decimals"

Total sweets = $1\frac{3}{4} = \frac{7}{4}$ kg. Number of friends = 7. Each friend receives $\frac{7}{4} \div 7 = \frac{7}{4} \times \frac{1}{7} = \frac{1}{4}$ kg.

Let the number be $x$. Then $\frac{3}{4} x = 12$. So $x = 12 \times \frac{4}{3} = 4 \times 4 = 16$.

The product of two fractions $\frac{a}{b}$ and $\frac{c}{d}$ is $\frac{a \times c}{b \times d}$. So, $\frac{2}{7} \times \frac{5}{9} = \frac{2 \times 5}{7 \times 9}$.

To make a fraction smallest, its numerator should be as small as possible and its denominator as large as possible. Given 0 < p < q < r < s, the sum $p+q$ is the smallest possible numerator among the options, and $r+s$ is the largest possible denominator. Therefore, $\frac{p+q}{r+s}$ will be the smallest.

The pattern shows each number is half of the previous number ($60 \div 2 = 30$, $30 \div 2 = 15$). So the next number is $15 \div 2 = \frac{15}{2}$.

To convert rupees and paise to rupees, remember that 1 rupee = 100 paise. So, 8 paise is equal to $8/100 = 0.08$ rupees. Adding this to 8 rupees gives $8 + 0.08 = 8.08$ rupees.

A regular hexagon has 6 equal sides. The perimeter is the sum of the lengths of all sides. Perimeter = $6 \times \text{side length} = 6 \times 3.5 \text{ cm} = 21 \text{ cm}$.

Dividing a number by 1000 means moving the decimal point 3 places to the left. So, $2.5 \div 1000 = 0.0025$.

Convert all options to decimal form for comparison: (a) 0.0002 (b) $\frac{2}{1000} = 0.002$ (c) $\frac{(0.2)^2}{2} = \frac{0.04}{2} = 0.02$ (d) $\frac{2}{100} \div 0.01 = 0.02 \div 0.01 = 2$ Comparing 0.0002, 0.002, 0.02, and 2, the smallest value is 0.0002.

Convert all options to decimal form for comparison: (a) $\frac{32}{0.05} = \frac{3200}{5} = 640$ (b) $\frac{0.320}{50} = \frac{0.32}{50} = 0.0064$ (c) $\frac{3.2}{0.05} = \frac{320}{5} = 64$ (d) $\frac{3.2}{50} = 0.064$ Comparing 640, 0.0064, 64, and 0.064, the largest value is 640.

Convert all options to decimal form for comparison: (a) 0.0001 (b) $\frac{1}{1000} = 0.001$ (c) $(0.100)^2 = (0.1)^2 = 0.01$ (d) $\frac{1}{10} \div 0.1 = 0.1 \div 0.1 = 1$ Comparing 0.0001, 0.001, 0.01, and 1, the largest value is 1.

In mathematics, particularly when dealing with fractions, the word 'of' acts as an operator that signifies multiplication. When we say 'a fraction acts as an operator of...', it implies that the fraction is used to multiply a quantity. For example, '1/2 of 10' means (1/2) × 10. The word 'of' in this context means to multiply. This concept is fundamental in understanding how fractions are applied to quantities, collections, or other numbers.

The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. Therefore, the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.

A proper fraction is less than 1, and an improper fraction is greater than 1. When you multiply a number by a value less than 1, the result is smaller than the original number. For example, $\frac{1}{2} \times \frac{3}{2} = \frac{3}{4}$. Here, $\frac{3}{4}$ is less than the improper fraction $\frac{3}{2}$.

Two non-zero fractions whose product is 1 are called reciprocals of each other. For example, $\frac{2}{3} \times \frac{3}{2} = 1$.

Since 1 rupee = 100 paise, 5 paise can be written as $\frac{5}{100}$ rupees, which is 0.05 rupees. So, 5 rupees 5 paise = 5 + 0.05 = 5.05 rupees.

There are 1000 millimeters (mm) in 1 meter (m). To convert millimeters to meters, divide by 1000. So, 45 mm = $\frac{45}{1000}$ m = 0.045 m.

To multiply a decimal number by 1000, move the decimal point three places to the right. $2.4 \times 1000 = 2400$.

When dividing a decimal number by 100, the decimal point is shifted two places to the left.

An improper fraction has a numerator greater than or equal to its denominator (e.g., $\frac{5}{3}$). Its reciprocal will have the denominator as the numerator and the numerator as the denominator (e.g., $\frac{3}{5}$). This reciprocal will be a proper fraction (less than 1), not an improper fraction.

First, convert the mixed fractions to improper fractions: $2\frac{2}{5} = \frac{(2 \times 5) + 2}{5} = \frac{12}{5}$ and $2\frac{1}{5} = \frac{(2 \times 5) + 1}{5} = \frac{11}{5}$. Now, perform the division: $\frac{12}{5} \div \frac{11}{5} = \frac{12}{5} \times \frac{5}{11} = \frac{12}{11}$. Since $\frac{12}{11} \neq 2$, the statement is false.