Percentage problems: a 10% increase, a 10% decrease, and the original price
After a 10% increase and a 10% decrease, a bicycle costs 1485, not 1500. After a 25% decrease, a sweatshirt costs 90, whereas before it cost 120. Three methods for the bicycle, four for the sweatshirt.
The lesson and video are in Polish. This is a translation of the description and lesson text for parents and students; Polish terms are given in brackets.
After you click, the video loads from YouTube (Google). The lesson is in Polish.
Lesson outline
Bicycle: price after a 10% increase and a 10% decrease
Step-by-step multipliers
A 10% increase means multiplying by 1,1, and a 10% decrease means multiplying by 0,9.
1500 · 1,1 = 1650; 1650 · 0,9 = 1485
Percentage of the price
We calculate the decrease from the new price, not from 1500.
10% of 1500 = 150, 1500 + 150 = 1650; 10% of 1650 = 165, 1650 − 165 = 1485
Single multiplier
Both changes together equal a single multiplication.
1,1 · 0,9 = 0,99, which is 99% of the price; 1500 · 0,99 = 1485
Sweatshirt: price before a 25% decrease
Equation
Let x be the price before the decrease. After a 25% decrease, 75% of the price remains.
x · 0,75 = 90, so x = 90 : 0,75 = 120
One percent
90 is 75% of the price. We calculate what 1% is and multiply by 100.
90 : 75 = 1,2; 1,2 · 100 = 120
Quarters
75% is three quarters of the price, so one quarter is 90 : 3.
90 : 3 = 30; 4 · 30 = 120
Proportion
75% is 90, and 100% is x. We cross-multiply.
x = 90 · 100 : 75 = 120
The most common mistake
Adding 25% to the reduced price: 90 + 22,5 = 112,5. But we calculate 25% of the old price, not 90. Check: 25% of 112,5 is 28,125, and 112,5 − 28,125 = 84,375 – not 90.
Your turn
After a 25% decrease, a pair of trousers costs 60. How much did they cost before the decrease?
Show the answer
80. 60 is 75% of the price; 1% = 60 : 75 = 0,8; 100% = 80. Check: 25% of 80 = 20, 80 − 20 = 60.
Download the worksheet (PDF, in Polish, with answers)
Lesson transcript
The lesson is recorded in Polish. Below is a translation of the lesson text; Polish terms are given in brackets.
Show the full lesson text
Introduction
A bicycle costs 1500. Its price was increased by 10% and then decreased by 10%. Has the price returned to 1500? In this part, we will solve 2 tasks like the ones on the Polish 8th-grade exam (egzamin ósmoklasisty). On the screen, you can see the outline and the minute when each task starts.
Recap
First, a reminder of what we did in part one. Pause the video and answer: what is the multiplier for a 20% increase? 1,2, because after the increase, the price is 120%. If you do not remember this, watch part one. The link is in the corner of the screen and in the description.
Task 1: bicycle – 3 methods
Before you calculate, write down three things. What we know: price 1500 + 10%, then minus 10%. What we are looking for: the final price. Task type: several percentage changes in a row. Method one: step-by-step multipliers. 1500 · 1,1 is 1650. Times 0,9 gives 1485. Method two: percentage of the price. 10% of 1500 is 150, so after the increase the bicycle costs 1650. Now pause the video and calculate yourself: how much is 10% of 1650? It is 165, which is more than 150. 1650 − 165 gives 1485. Method three: a single multiplier. 1,1 · 0,9 is 0,99, which is 99%. The price dropped by 1%, which is 15. Three methods and the same result: 1485. The price has not returned. The order of changes does not matter: first a decrease and then an increase give the same result.
Task 2: sweatshirt – 4 methods
The second task is the opposite. After a 25% decrease, a sweatshirt costs 90. How much did it cost before the decrease? Pay attention to one word. A 25% decrease means the price decreased by one quarter. That is not the same as a decrease down to 25%. Pause the video and choose the answer. A: 112,5. B: 120. C: 115. Answer A is the most common mistake. It happens when someone adds 25% to 90. But we calculate 25% of the old price, not 90. Before you calculate: a price of 90 is what remains after the decrease, which is 75% of the old price. We are looking for 100%. Task type: price before a change. I will show you four methods. Choose the one that is clearest to you. Method one: equation. We denote the price before the decrease with the letter x. x times 0,75 = 90. We divide 90 by 0,75 and we get 120. Give it a try. Method two: by 1%. 75 squares (kratki) is 90, so one square (kratka) is 1,2. 100 squares (kratki) is 120. Method three: quarters. 75% is 3/4. 90 is 3 quarters, so one quarter is 30, and 4 quarters is 120. Method four: proportion. 75% is 90, and 100% is x. We cross-multiply: x equals 90 · 100, divided by 75, which is 120. Let us check: 25% of 120 is 30. 120 − 30 is 90. The result is correct.
When to use which method
When to use which method? Quarters are the fastest when the percentage is 25, 50, or 75. The 1% method always works. The equation and proportion also always work and are easiest to write down on the exam. Steps for such a task. One: how much percentage is left. 2: how much is 1%. 3: times 100. 4: check the result.
Your turn and questions
Your turn. After a 25% decrease, a pair of trousers costs 60. How much did they cost before the decrease? Write the result in the comments. Below the video, you will find a worksheet with answers. Each of the four methods is also in a separate short video, links are in the description. If anything was unclear, ask questions in the comments. We will answer every single one.