Ratio and Proportion Cheatshee

Cheatsheet Content

### 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 means (wells, canals, purchased water). **Example:** A farmer harvests 1000 kg of wheat from naturally irrigated land. Ushr amount = $1000 \text{ kg} \times 10\% = 1000 \times 0.10 = 100 \text{ kg}$. The farmer must give 100 kg of wheat as Ushr. #### Inheritance and its Ratio of Share Between Family In Islamic law, inheritance (Mawarith) is distributed according to specific ratios outlined in the Quran and Sunnah. These ratios vary depending on the number and relationship of surviving heirs. Common shares for fixed heirs include: - **Wife:** 1/8 if there are children, 1/4 if no children. - **Husband:** 1/4 if there are children, 1/2 if no children. - **Daughter(s):** 1/2 for one daughter, 2/3 for two or more daughters (if no son). - **Son(s):** Sons inherit as residuaries, usually receiving double the share of a daughter (if both are present). **Example (Simplified):** A man dies leaving behind a wife, one son, and one daughter. His estate is $120,000. 1. **Wife's Share:** Wife gets 1/8 because there are children. Share = $120,000 \times \frac{1}{8} = $15,000$. 2. **Remaining Estate:** $120,000 - $15,000 = $105,000$. 3. **Son & Daughter Share:** The remaining estate is divided between son and daughter in a $2:1$ ratio. Let son's share be $2x$ and daughter's share be $x$. $2x + x = 105,000$ $3x = 105,000$ $x = 35,000$ - **Daughter's Share:** $x = $35,000$. - **Son's Share:** $2x = 2 \times 35,000 = $70,000$. Final distribution: Wife $15,000, Daughter $35,000, Son $70,000. ### Profit and Loss #### Cost Price (CP) The price at which an article is purchased. #### Selling Price (SP) The price at which an article is sold. #### Profit When SP > CP. **Profit = SP - CP** #### Loss When CP > SP. **Loss = CP - SP** #### Profit Percentage $\text{Profit Percentage} = \frac{\text{Profit}}{\text{CP}} \times 100\%$ #### Loss Percentage $\text{Loss Percentage} = \frac{\text{Loss}}{\text{CP}} \times 100\%$ **Example:** A shopkeeper buys an item for $500 and sells it for $600. CP = $500 SP = $600 Profit = SP - CP = $600 - $500 = $100$. Profit Percentage = $\frac{100}{500} \times 100\% = \frac{1}{5} \times 100\% = 20\%$. If the shopkeeper had sold it for $450 instead: CP = $500 SP = $450 Loss = CP - SP = $500 - $450 = $50$. Loss Percentage = $\frac{50}{500} \times 100\% = \frac{1}{10} \times 100\% = 10\%$. ### Discount and Discount Percentage #### Marked Price (MP) / List Price The price at which an item is listed for sale, often displayed on the tag. #### Discount A reduction offered on the Marked Price. **Discount = MP - SP** #### Discount Percentage $\text{Discount Percentage} = \frac{\text{Discount}}{\text{MP}} \times 100\%$ **Example:** A shirt is marked at $800. The shop offers a 15% discount. MP = $800 Discount Percentage = 15% Discount Amount = $800 \times 15\% = $800 \times \frac{15}{100} = $120$. Selling Price (SP) = MP - Discount Amount = $800 - $120 = $680$. To find the discount percentage if an item marked $1200 is sold for $1000: MP = $1200 SP = $1000 Discount = MP - SP = $1200 - $1000 = $200$. Discount Percentage = $\frac{200}{1200} \times 100\% = \frac{1}{6} \times 100\% \approx 16.67\%$. ### Business Partnership A **business partnership** involves two or more individuals who agree to share in the profits or losses of a business venture. Profits and losses are usually shared in the ratio of their investments and the time period for which the investment was made. #### Ratio of Profits/Losses If partners $P_1, P_2, \dots, P_n$ invest amounts $A_1, A_2, \dots, A_n$ for time periods $T_1, T_2, \dots, T_n$ respectively, then their share of profit or loss will be in the ratio: $P_1 : P_2 : \dots : P_n = (A_1 \times T_1) : (A_2 \times T_2) : \dots : (A_n \times T_n)$ **Example:** A, B, and C start a business. A invests $10,000 for 12 months, B invests $12,000 for 8 months, and C invests $15,000 for 6 months. If the total profit at the end of the year is $16,200, find each partner's share. Ratio of investments $\times$ time: A: $10,000 \times 12 = 120,000$ B: $12,000 \times 8 = 96,000$ C: $15,000 \times 6 = 90,000$ Ratio of profits A : B : C = $120,000 : 96,000 : 90,000$ Divide by 1000: $120 : 96 : 90$ Divide by 6: $20 : 16 : 15$ Sum of ratios = $20 + 16 + 15 = 51$. Total Profit = $16,200. A's share = $\frac{20}{51} \times 16,200 = 20 \times 317.65 \approx $6352.94$ B's share = $\frac{16}{51} \times 16,200 = 16 \times 317.65 \approx $5082.35$ C's share = $\frac{15}{51} \times 16,200 = 15 \times 317.65 \approx $4764.71$ (Note: Due to rounding, the sum might not be exactly $16,200.) Let's use exact division for more precise results: $16200 / 51 = 317.6470588...$ A's share = $20 \times (16200/51) = 324000/51 = $6352.94$ B's share = $16 \times (16200/51) = 259200/51 = $5082.35$ C's share = $15 \times (16200/51) = 243000/51 = $4764.71$ Sum = $6352.94 + 5082.35 + 4764.71 = 16200.00$