Hi there! Today, I’m excited to share with you an essential part of mathematics: percentages and how they work. You’ve likely learned the basics back in primary school, such as what percentages mean and how to convert between percentages, decimals, and fractions. Maybe you even solved simple problems involving percentages and percentage changes. But as we move forward, we’re going to dive deeper and unlock the power of percentages in more complex situations.
Let me start by explaining percentage change. Imagine you’re looking at how a value increases or decreases—this is where percentage change becomes super useful. Let’s take a quick example: if something grows from 100 to 120, the increase is 20% (also called a percentage change of +20%). But be careful! If that same thing shrinks from 120 back to 100, the decrease isn’t 20%. Why? It’s because percentage change depends on the original value. That’s why we need to understand the relationship between the original value, new value, and percentage change. This concept is key to mastering percentage problems.
Here’s a quick formula to remember:
- New value = Original value × (1 + Percentage change)
- Percentage change = ((New value − Original value) ÷ Original value) × 100%
Keep in mind, the percentage change can be positive (an increase) or negative (a decrease). Learning how this applies to real-world problems makes math so much more practical—and honestly, fun!
Next, we’ll use these concepts in real-life scenarios. For example:
- Profit and loss problems: You’ll learn terms like cost price, selling price, and marked price. Imagine a product being sold at a discount—terms like “20% off” or “85% of the price” will make complete sense, and you’ll know exactly how to calculate the effects.
- Simple and compound interest: When we talk about savings accounts or loans, we’ll explore terms like principal, interest rate, and duration. You’ll also figure out the difference between simple and compound interest.
- Taxes and depreciation: You’ll see how percentages are used to calculate salary taxes or how machinery loses its value over time. These are practical uses of math in daily life.
Here’s a fun fact: you can even explore how percentages describe things in science or technology, like the growth rate of a species in nature or the depreciation rate of machine parts. Knowing percentages lets us describe these changes clearly and understand them better.
So, as you work through percentage problems, remember to use the formulas flexibly. Instead of memorizing tons of different formulas, focus on applying these two formulas in various situations. By the end of this lesson, you’ll see just how powerful percentages are—not only in math class but also in understanding the world around you. Let’s dive in and master it together!
Example Problems and Solutions in Percentages
Problem 1 The marked price of a product is $200, and a 25% discount is offered. (a) What is the selling price of the product? (b) If the product’s cost price is $140, calculate the profit percentage after selling.
Solution (a) Discount Amount = Marked price × Discount percentage = $200 × 25% = $200 × 0.25 = $50 Selling Price = Marked price − Discount Amount = $200 − $50 = $150
(b) Profit = Selling price − Cost price = $150 − $140 = $10 Profit Percentage = (Profit ÷ Cost price) × 100% = ($10 ÷ $140) × 100% ≈ 7.14%
Problem 2 The cost price of an item is $500, and it is sold for $650. (a) Calculate the profit percentage. (b) If the item’s original marked price was $750, determine the discount percentage offered.
Solution (a) Profit = Selling price − Cost price = $650 − $500 = $150 Profit Percentage = (Profit ÷ Cost price) × 100% = ($150 ÷ $500) × 100% = 30%
(b) Discount Amount = Marked price − Selling price = $750 − $650 = $100 Discount Percentage = (Discount Amount ÷ Marked price) × 100% = ($100 ÷ $750) × 100% ≈ 13.33%
Extended Learning The above examples show the use of percentage calculations in practical scenarios. Once you are comfortable with these problems, you can challenge yourself further with more complex topics, such as:
- Tax Calculations: For instance, determining the total price of a product after adding tax, or calculating the pre-tax amount from the final price.
- Interest Problems: Compare simple and compound interest, and learn about their relationship with principal, rate, and time.
- Depreciation and Growth Rates: Explore how the value of an asset decreases over time, or calculate the growth rate of an investment or population.
Practicing these advanced problems will help you apply the percentage formulas more flexibly and understand their significance in real-life situations. Keep challenging yourself!
Below are some practice problems involving the application of percentages.
結果
#1. The cost of a pen is $500, its marked price is $600, and it is sold for $540. (a) Find the discount percentage. (b) Calculate the profit or loss percentage.
#2. The cost price and marked price of a mobile phone are $3500 and $5200, respectively. If the mobile phone is sold at a 30% discount, will the seller incur a loss?
#3. The marked price of a bottle of correction fluid is $16. During a promotional offer, a discount of $4 is given for each bottle purchased. If the cost price of each bottle is $10, calculate the profit or loss percentage.
我本業其實是做日文翻譯、攝影、還有設計，這三樣加起來，離數學差不多有一百萬公里遠。本來以為這輩子都不會再被 x、y、指數、小括號纏住。結果女兒一入學，家中最高領導人一聲令下：「數學交給你了。」
我當場愣住，心想我以前看到代數都頭痛，現在竟然要主管全家的算式業務？每天還要追蹤進度、檢查功課、臨場講解，完全比我接案子還拼命。
既然跑不掉，那乾脆把平日教女兒時累積的一堆小抄、筆記全整理起來。反正我自己要用，順便讓世上其他同樣被指派「數學專員」的苦主也能受惠。能救一個是一個，能救全家更好。