}]]}]}]}] 30:T2405,

### Ratio A **ratio** is a comparison of two or more quantities of the same kind, expressed as $a:b$ or $\frac{a}{b}$. It shows how many times one quantity contains another. #### Ratio as a Relation A ratio establishes a relationship between two numbers, indicating their relative sizes. For example, if there are 3 apples and 2 oranges, the ratio of apples to oranges is $3:2$. #### Antecedent and Consequent In a ratio $a:b$: - **Antecedent:** The first term, $a$. - **Consequent:** The second term, $b$. #### Increase and Decrease in Ratio To increase or decrease a quantity in a given ratio $a:b$: - **New Quantity** = Original Quantity $\times \frac{b}{a}$ **Example:** Increase 60 in the ratio $3:2$. Original Quantity = 60 Ratio = $3:2$ (Meaning the new quantity will be larger than the original) New Quantity = $60 \times \frac{3}{2} = 30 \times 3 = 90$ Decrease 90 in the ratio $2:3$. Original Quantity = 90 Ratio = $2:3$ (Meaning the new quantity will be smaller than the original) New Quantity = $90 \times \frac{2}{3} = 30 \times 2 = 60$ ### Proportion A **proportion** is an equality of two ratios. If $a:b = c:d$, then $a, b, c, d$ are said to be in proportion. This can also be written as $\frac{a}{b} = \frac{c}{d}$. #### Extremes and Means In a proportion $a:b = c:d$: - **Extremes:** The first and fourth terms ($a$ and $d$). - **Means:** The second and third terms ($b$ and $c$). A fundamental property of proportion is that the product of the extremes equals the product of the means: $a \times d = b \times c$. #### Direct Proportion Two quantities are in **direct proportion** if an increase in one quantity causes a proportional increase in the other, and vice versa. If $y \propto x$, then $y = kx$ for some constant $k$. **Example:** If 3 pens cost $15, how much do 7 pens cost? Let $x$ be the cost of 7 pens. $$\frac{3 \text{ pens}}{15 \text{ dollars}} = \frac{7 \text{ pens}}{x \text{ dollars}}$$ $$3x = 7 \times 15$$ $$3x = 105$$ $$x = \frac{105}{3} = 35$$ So, 7 pens cost $35. #### Inverse Proportion Two quantities are in **inverse proportion** if an increase in one quantity causes a proportional decrease in the other, and vice versa. If $y \propto \frac{1}{x}$, then $y = \frac{k}{x}$ for some constant $k$. **Example:** If 4 workers can complete a task in 10 days, how many days will 8 workers take to complete the same task? Let $d$ be the number of days for 8 workers. (More workers mean less days, so it's inverse proportion) $$4 \text{ workers} \times 10 \text{ days} = 8 \text{ workers} \times d \text{ days}$$ $$40 = 8d$$ $$d = \frac{40}{8} = 5$$ So, 8 workers will take 5 days. ### Proportion: Real-Life Problems #### Zakat **Zakat** is an obligatory annual charity in Islam, paid by Muslims who meet the necessary criteria of wealth. It is a form of social welfare. - **Nisab of Zakat:** The minimum amount of wealth a Muslim must possess before they are liable to pay Zakat. - For gold, it is 87.48 grams (or 7.5 tolas) of pure gold. - For silver, it is 612.36 grams (or 52.5 tolas) of pure silver. - Equivalent values apply to cash, savings, and tradable assets. - **Rate of Zakat:** 2.5% of the total accumulated wealth (above Nisab) that has been in one's possession for a full lunar year. **Example:** A person has $10,000 in savings for a full lunar year, and the Nisab for cash is less than this amount. Zakat amount = $10,000 \times 2.5\% = 10,000 \times \frac{2.5}{100} = 10,000 \times 0.025 = $250$. The person must pay $250 as Zakat. #### Ushr **Ushr** is Zakat on agricultural produce. - **Rate of Ushr:** - 10% of the produce if the land is irrigated by natural means (rain, springs). - 5% of the produce if the land is irrigated by artificial mean