Mastering Percentages for SAT Math Success
Introduction
Percentage problems are a common feature of the SAT Math section. They test your ability to work with ratios, proportions, and changes in quantities. Whether it’s calculating a discount, tax, or percentage increase, knowing these problems inside and out can help you gain easy points.
This guide will cover:
- Key formulas for percentage calculations.
- Solving step-by-step percentage problems.
- Common SAT percentage question types.
Key Percentage Formulas
Percent means “per hundred.” The basic formula is:
![\[ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMzYiIGhlaWdodD0iNDMiIHZpZXdCb3g9IjAgMCAyMzYgNDMiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
Percentage Increase/Decrease
![\[ \text{Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzNTIiIGhlaWdodD0iNDAiIHZpZXdCb3g9IjAgMCAzNTIgNDAiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
Finding the Whole Given the Part
![\[ \text{Whole} = \frac{\text{Part}}{\text{Percentage}} \times 100 \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTAiIGhlaWdodD0iNDEiIHZpZXdCb3g9IjAgMCAyMTAgNDEiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
![\[ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-543f22020a3386ee44130d9f07b12d3c_l3.png "Rendered by QuickLaTeX.com")
![\[ \text{Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-8eb8c853985de4f505f9eb60dd829c4c_l3.png "Rendered by QuickLaTeX.com")
![\[ \text{Whole} = \frac{\text{Part}}{\text{Percentage}} \times 100 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-7d4e4638adfa281cb168b9b0c5d6b19c_l3.png "Rendered by QuickLaTeX.com")
Solving Percentage Problems Step-by-Step
Example 1: Basic Percentage Calculation
Question: What is 20% of 150?
Solution:
![\[ \text{Percentage} = \left( \frac{20}{100} \right) \times 150 = 30 \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNTQiIGhlaWdodD0iNDMiIHZpZXdCb3g9IjAgMCAyNTQgNDMiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
Example 2: Percentage Increase
Question: A jacket costs $80 and the price increases by 25%. What is the new price?
Solution:
- Find the increase:
![\[ \text{Increase} = \left( \frac{25}{100} \right) \times 80 = 20 \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMjQiIGhlaWdodD0iNDMiIHZpZXdCb3g9IjAgMCAyMjQgNDMiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
- Add to original price:
![\[ \text{New Price} = 80 + 20 = 100 \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTEiIGhlaWdodD0iMTQiIHZpZXdCb3g9IjAgMCAyMTEgMTQiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
Example 3: Percentage Decrease
Question: A phone is discounted by 30% from $200. What is the sale price?
Solution:
- Find the discount:
![\[ \text{Discount} = \left( \frac{30}{100} \right) \times 200 = 60 \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMzgiIGhlaWdodD0iNDMiIHZpZXdCb3g9IjAgMCAyMzggNDMiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
- Subtract from original price:
![\[ \text{Sale Price} = 200 - 60 = 140 \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTgiIGhlaWdodD0iMTMiIHZpZXdCb3g9IjAgMCAyMTggMTMiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
![\[ \text{Percentage} = \left( \frac{20}{100} \right) \times 150 = 30 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-5bf45723ae19b696efa5875d356a79e0_l3.png "Rendered by QuickLaTeX.com")
![\[ \text{Increase} = \left( \frac{25}{100} \right) \times 80 = 20 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-15a6b5d4f8fa210426ec6e9fd75be46c_l3.png "Rendered by QuickLaTeX.com")
![\[ \text{New Price} = 80 + 20 = 100 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-c2edb29bb4abc6b88d485208312be03b_l3.png "Rendered by QuickLaTeX.com")
![\[ \text{Discount} = \left( \frac{30}{100} \right) \times 200 = 60 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-abb5f6062f43c57f141a1db2bb538ddf_l3.png "Rendered by QuickLaTeX.com")
![\[ \text{Sale Price} = 200 - 60 = 140 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-17fec35f64f211a67c300081a552e5b3_l3.png "Rendered by QuickLaTeX.com")
SAT Percentage Problem Types
Finding the Percentage of a Number
- Use the basic percentage formula:
.
Percentage Increase and Decrease
- Always compare the change relative to the original value.
Reverse Percentage Problems
- Find the original price or value before a percentage increase or decrease.
- Use the basic percentage formula:
.Practice Question
Question: The price of a shirt increased from $50 to $65. What is the percentage increase?
Solution:
- Find the change:
![\[ \text{Change} = 65 - 50 = 15 \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzkiIGhlaWdodD0iMTciIHZpZXdCb3g9IjAgMCAxNzkgMTciPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
- Divide by the original value:
![\[ \text{Percentage Increase} = \frac{15}{50} \times 100 = 30\% \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzMDEiIGhlaWdodD0iMzciIHZpZXdCb3g9IjAgMCAzMDEgMzciPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
Answer: The percentage increase is 30%.
![\[ \text{Change} = 65 - 50 = 15 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-eff4676eee47d5f7599dd1015ba511b7_l3.png "Rendered by QuickLaTeX.com")
![\[ \text{Percentage Increase} = \frac{15}{50} \times 100 = 30\% \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-3752721e8c16cd47a76e7c3c8c046e4e_l3.png "Rendered by QuickLaTeX.com")
Common Mistakes to Avoid
- Confusing Increase/Decrease: Always compare to the original value.
- Skipping Steps: Write out each step to avoid calculation errors.
- Misreading “of” vs. “off”: “20% of 100” means multiplication; “20% off 100” involves subtraction.
Summary
Mastering percentages for the SAT requires understanding key formulas and problem types. With practice, you’ll solve these questions quickly and accurately to boost your score.
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