Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 6: Ratios and Percents

Lesson 5: Percentage Problems

In this Grade 4 AMC Math lesson from AoPS Introduction to Algebra, students tackle percentage problems including percent increase and decrease using the formulas x(1 + k/100) and x(1 - k/100). The lesson covers converting between percents, decimals, and fractions, and emphasizes a critical concept: successive percentage changes multiply rather than add. Students also practice choosing convenient values for unknown quantities to simplify ratio and percent calculations.

Section 1

Percentage increase and decrease formulas

Property

To find a percentage of a number, multiply the number by the decimal equivalent of the percent. For example, 15% of 200 is 0.15×200=300.15 \times 200 = 30. To increase a number by a percentage, multiply by (1+decimal)(1 + \text{decimal}). To decrease a number by a percentage, multiply by (1−decimal)(1 - \text{decimal}). For example, an increase of 8% means multiplying by 1.08, and a decrease of 8% means multiplying by 0.92.

Examples

Section 2

Multiplying Sequential Percentage Change Factors

Property

For sequential percentage changes, multiply the change factors: (1±p1100)×(1±p2100)×…(1 \pm \frac{p_1}{100}) \times (1 \pm \frac{p_2}{100}) \times \ldots

Do NOT add the percentages: p1+p2+…p_1 + p_2 + \ldots (this is incorrect)

Section 3

Understanding Asymmetric Percentage Changes

Property

Percentage increases and decreases are not symmetric operations. A kk% increase followed by a kk% decrease does not return to the original value: x(1+k100)(1−k100)=x(1−k210000)≠xx(1 + \frac{k}{100})(1 - \frac{k}{100}) = x(1 - \frac{k^2}{10000}) \neq x

Examples

Section 4

Calculating Weighted Percentages

Property

To find the overall percentage for a population composed of different groups, calculate the weighted average. The formula for groups with sizes N1,N2,...,NkN_1, N_2, ..., N_k and corresponding percentages P1,P2,...,PkP_1, P_2, ..., P_k is:

Overall Percentage=(N1×P1)+(N2×P2)+...+(Nk×Pk)N1+N2+...+Nk \text{Overall Percentage} = \frac{(N_1 \times P_1) + (N_2 \times P_2) + ... + (N_k \times P_k)}{N_1 + N_2 + ... + N_k}

Examples

  • At a conference, 60% of the attendees are men and 40% are women. If 50% of the men and 70% of the women are from out of state, the percentage of all attendees from out of state is: (60×0.50)+(40×0.70)60+40=30+28100=58%\frac{(60 \times 0.50) + (40 \times 0.70)}{60 + 40} = \frac{30 + 28}{100} = 58\%.
  • A school has twice as many students in Grade 10 as in Grade 11. If 80% of Grade 10 students and 95% of Grade 11 students have a laptop, the percentage of students in both grades who have a laptop is: (2×0.80)+(1×0.95)2+1=1.60+0.953=2.553=85%\frac{(2 \times 0.80) + (1 \times 0.95)}{2 + 1} = \frac{1.60 + 0.95}{3} = \frac{2.55}{3} = 85\%.

Explanation

A weighted percentage is an average that accounts for the different sizes of the subgroups being considered. Instead of simply averaging the percentages, you must "weigh" each percentage by the size or proportion of its group. To calculate it, multiply each group''s percentage by its size (or relative weight), sum these products, and then divide by the total size of all groups combined. This method is crucial when subgroups are of unequal size, as it gives a more accurate overall picture.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Percentage increase and decrease formulas

Property

To find a percentage of a number, multiply the number by the decimal equivalent of the percent. For example, 15% of 200 is 0.15×200=300.15 \times 200 = 30. To increase a number by a percentage, multiply by (1+decimal)(1 + \text{decimal}). To decrease a number by a percentage, multiply by (1−decimal)(1 - \text{decimal}). For example, an increase of 8% means multiplying by 1.08, and a decrease of 8% means multiplying by 0.92.

Examples

Section 2

Multiplying Sequential Percentage Change Factors

Property

For sequential percentage changes, multiply the change factors: (1±p1100)×(1±p2100)×…(1 \pm \frac{p_1}{100}) \times (1 \pm \frac{p_2}{100}) \times \ldots

Do NOT add the percentages: p1+p2+…p_1 + p_2 + \ldots (this is incorrect)

Section 3

Understanding Asymmetric Percentage Changes

Property

Percentage increases and decreases are not symmetric operations. A kk% increase followed by a kk% decrease does not return to the original value: x(1+k100)(1−k100)=x(1−k210000)≠xx(1 + \frac{k}{100})(1 - \frac{k}{100}) = x(1 - \frac{k^2}{10000}) \neq x

Examples

Section 4

Calculating Weighted Percentages

Property

To find the overall percentage for a population composed of different groups, calculate the weighted average. The formula for groups with sizes N1,N2,...,NkN_1, N_2, ..., N_k and corresponding percentages P1,P2,...,PkP_1, P_2, ..., P_k is:

Overall Percentage=(N1×P1)+(N2×P2)+...+(Nk×Pk)N1+N2+...+Nk \text{Overall Percentage} = \frac{(N_1 \times P_1) + (N_2 \times P_2) + ... + (N_k \times P_k)}{N_1 + N_2 + ... + N_k}

Examples

  • At a conference, 60% of the attendees are men and 40% are women. If 50% of the men and 70% of the women are from out of state, the percentage of all attendees from out of state is: (60×0.50)+(40×0.70)60+40=30+28100=58%\frac{(60 \times 0.50) + (40 \times 0.70)}{60 + 40} = \frac{30 + 28}{100} = 58\%.
  • A school has twice as many students in Grade 10 as in Grade 11. If 80% of Grade 10 students and 95% of Grade 11 students have a laptop, the percentage of students in both grades who have a laptop is: (2×0.80)+(1×0.95)2+1=1.60+0.953=2.553=85%\frac{(2 \times 0.80) + (1 \times 0.95)}{2 + 1} = \frac{1.60 + 0.95}{3} = \frac{2.55}{3} = 85\%.

Explanation

A weighted percentage is an average that accounts for the different sizes of the subgroups being considered. Instead of simply averaging the percentages, you must "weigh" each percentage by the size or proportion of its group. To calculate it, multiply each group''s percentage by its size (or relative weight), sum these products, and then divide by the total size of all groups combined. This method is crucial when subgroups are of unequal size, as it gives a more accurate overall picture.