4  Percent change

4.1 Absolute and percent change

According to the Spring 2026 Census Snapshot, in Spring 2025 UNM-Taos had 881 home students and in Spring 2026, there were 933 home students.

The absolute change in the number of home students is computed by taking the ending value and subtracting the starting value: \[ 933 - 881 = 52\text{ students}. \]

The percent change in the number of home students is computed by viewing the absolute change as a proportion of the starting number: \[ \frac{52\text{ students}}{881\text{ students}} = 0.059 = 5.9%\text{ increase}. \]

Example 4.1 (Practice: absolute and percent change)  

  1. In 2010, Taos County had a population of 32,937. In 2020, the population was 34,489. Compute the absolute change and the percent change.

  2. This year, Los Luceros Historic Site has 51 Churro sheep. Last year, they had 43 sheep. Compute the absolute change and the percent change.

  3. The Town of Taos had a population of 5,716 in 2010. In 2020, the population was 6,474. Compute the absolute change and the percent change.

  4. This week the UNM Taos Cafe sold 23 frito pies. Last week they sold 37 frito pies. Compute the absolute change and the percent change.

  1. Absolute change: \(34,489 - 32,937 = 1,552\) people

    Percent change: \(\dfrac{1,552}{32,937} \approx 0.0471 = 4.71\%\) increase

  2. Absolute change: \(51 - 43 = 8\) sheep

    Percent change: \(\dfrac{8}{43} \approx 0.1860 = 18.60\%\) increase

  3. Absolute change: \(6,474 - 5,716 = 758\) people

    Percent change: \(\dfrac{758}{5,716} \approx 0.1326 = 13.26\%\) increase

  4. Absolute change: \(23 - 37 = -14\) frito pies

    Percent change: \(\dfrac{-14}{37} \approx -0.3784 = -37.84\%\) decrease

4.2 Percent growth

A local credit union is offering 1-year certificates of deposit (CDs) with an annual precentage rate of 2.5%. This means that if we deposit $500 in a CD account now, the value in one year will be 2.5% higher.

One way to compute the value after one year is to first compute the change in the value: \[ \text{change in value} = 2.5\%\text{ of }\$500 = 0.025\cdot 500 = \$12.50. \] Thus after one year we have \[ \begin{aligned} \text{new value} &= \text{old value} + \text{change in value} \\ &= \$500 + \$12.50 \\ &= \$512.50. \end{aligned} \]

A second way to compute the value after one year is to combine the percents: \[ \begin{aligned} \text{new value} &= 100\%\text{ of the original amount} + 2.5\%\text{ of the original amount} \\ &= 102.5\%\text{ of the original amount}. \end{aligned} \] This means that the new value is equal to \[ 102.5\%\text{ of }\$500 = 1.025\cdot 500 = \$512.50. \]

While both methods are equally valid, you are encouraged to use the second method wherever possible.

Example 4.2 (Practice: percent growth)  

  1. The Town of Taos currently has a population of 6,474. Demographers project a 5% increase over the next decade. What will the population be after this increase?

  2. A rancher purchases a truck for $45,000. The truck is expected to lose 12% of its value during the first year. What will the truck’s value be after one year?

  3. UNM-Taos currently has 1,321 students enrolled. The administration hopes for an 8% increase in enrollment next year. If this happens, how many students will be enrolled?

  4. A small business in Taos had revenue of $125,000 last year. Due to economic conditions, they expect a 6% decrease in revenue this year. What will their revenue be after this decrease?

  1. \(6,474 \cdot 1.05 = 6,797.7 \approx 6,798\) people

  2. \(45,000 \cdot 0.88 = 39,600\) dollars

  3. \(1,321 \cdot 1.08 = 1,426.68 \approx 1,427\) students

  4. \(125,000 \cdot 0.94 = 117,500\) dollars

4.3 Homework exercises

Exercise 4.1 For each scenario, compute the absolute change and the percent change. Round percent changes to two decimal places.

  1. In 2010, New Mexico had a population of 2,059,179. In 2020, the population was 2,117,522. Compute the absolute change and the percent change.

  2. Bernalillo County had a population of 662,564 in 2010. In 2020, the population was 676,444. Compute the absolute change and the percent change.

  3. This year, a Taos gallery sold 145 pieces of artwork. Last year, they sold 128 pieces. Compute the absolute change and the percent change.

  4. In 2020, Santa Fe County had a population of 154,823. In 2010, the population was 144,170. Compute the absolute change and the percent change.

  5. A local farm stand sold 89 pounds of green chile this week. Last week they sold 112 pounds. Compute the absolute change and the percent change.

  6. Rio Arriba County had a population of 40,246 in 2010. In 2020, the population was 40,363. Compute the absolute change and the percent change.

  7. The Taos Ski Valley reported 3,250 skier visits this month. Last month they had 2,890 visits. Compute the absolute change and the percent change.

  8. Doña Ana County had a population of 209,233 in 2010. In 2020, the population was 218,195. Compute the absolute change and the percent change.

  1. Absolute change: \(2,117,522 - 2,059,179 = 58,343\) people

    Percent change: \(\dfrac{58,343}{2,059,179} \approx 0.0283 = 2.83\%\) increase

  2. Absolute change: \(676,444 - 662,564 = 13,880\) people

    Percent change: \(\dfrac{13,880}{662,564} \approx 0.0209 = 2.09\%\) increase

  3. Absolute change: \(145 - 128 = 17\) pieces

    Percent change: \(\dfrac{17}{128} \approx 0.1328 = 13.28\%\) increase

  4. Absolute change: \(154,823 - 144,170 = 10,653\) people

    Percent change: \(\dfrac{10,653}{144,170} \approx 0.0739 = 7.39\%\) increase

  5. Absolute change: \(89 - 112 = -23\) pounds

    Percent change: \(\dfrac{-23}{112} \approx -0.2054 = -20.54\%\) decrease

  6. Absolute change: \(40,363 - 40,246 = 117\) people

    Percent change: \(\dfrac{117}{40,246} \approx 0.0029 = 0.29\%\) increase

  7. Absolute change: \(3,250 - 2,890 = 360\) visits

    Percent change: \(\dfrac{360}{2,890} \approx 0.1246 = 12.46\%\) increase

  8. Absolute change: \(218,195 - 209,233 = 8,962\) people

    Percent change: \(\dfrac{8,962}{209,233} \approx 0.0428 = 4.28\%\) increase

Exercise 4.2 For each scenario, compute the resulting value after the percent change. Round to the nearest whole number when appropriate.

  1. Taos County currently has a population of 34,489. Demographers project a 3% increase over the next decade. What will the population be after this increase?

  2. A restaurant in Taos purchases kitchen equipment for $28,000. The equipment is expected to lose 15% of its value during the first year. What will the equipment’s value be after one year?

  3. A local nonprofit had 450 volunteers last year. They hope for a 10% increase in volunteers this year. If this happens, how many volunteers will they have?

  4. Los Luceros Historic Site had 8,500 visitors last year. Due to road construction, they expect a 7% decrease in visitors this year. What will their visitor count be after this decrease?

  5. The median household income in Taos County is $58,950. Economists project a 4% increase over the next five years. What will the median household income be after this increase?

  6. A Taos art gallery had sales of $185,000 last year. Due to increased tourism, they expect a 12% increase in sales this year. What will their sales be after this increase?

  7. A sheep ranch currently has 240 sheep. Due to drought conditions, the rancher expects an 8% decrease in the herd this year. How many sheep will remain after this decrease?

  8. New Mexico had a population of 2,117,522 in 2020. If the population grows by 2% over the next decade, what will the population be?

  1. \(34,489 \cdot 1.03 = 35,523.67 \approx 35,524\) people

  2. \(28,000 \cdot 0.85 = 23,800\) dollars

  3. \(450 \cdot 1.10 = 495\) volunteers

  4. \(8,500 \cdot 0.93 = 7,905\) visitors

  5. \(58,950 \cdot 1.04 = 61,308\) dollars

  6. \(185,000 \cdot 1.12 = 207,200\) dollars

  7. \(240 \cdot 0.92 = 220.8 \approx 221\) sheep

  8. \(2,117,522 \cdot 1.02 = 2,159,872.44 \approx 2,159,872\) people