What is a Word Problem on AP? | Simple Explanation for Class 10

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What is a Word Problem on Arithmetic Progressions?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A word problem on Arithmetic Progressions (AP) is a story-based math problem where quantities increase or decrease by a constant amount each time. You need to identify the first term, common difference, and apply AP formulas to find missing values like a specific term or the sum of terms.

Simple Example

Quick Example

Imagine you save Rs 50 on the first day, Rs 60 on the second day, Rs 70 on the third day, and so on. This is an AP because your savings increase by a constant Rs 10 each day. A word problem might ask how much you save on the 10th day.

Worked Example

Step-by-Step

PROBLEM: An auto-rickshaw driver charges Rs 30 for the first kilometer and Rs 8 for every additional kilometer. If a passenger travels 12 kilometers, what is the total fare?

STEP 1: Identify the first term (a). The charge for the first kilometer is Rs 30. So, a = 30.
---STEP 2: Identify the common difference (d). The charge for every additional kilometer is Rs 8. So, d = 8.
---STEP 3: Understand what needs to be found. The passenger travels 12 kilometers. We need the total fare, which is the sum of charges for 12 kilometers. This is a sum of an AP, but the first term is for 1km, and subsequent 'terms' are for additional kilometers. Let's reframe: The first term is Rs 30. The remaining (12-1) = 11 kilometers will each cost Rs 8.
---STEP 4: Calculate the cost for the additional kilometers. 11 kilometers * Rs 8/kilometer = Rs 88.
---STEP 5: Calculate the total fare. Total fare = Cost for 1st km + Cost for additional kilometers.
---STEP 6: Total fare = Rs 30 + Rs 88 = Rs 118.
ANSWER: The total fare for 12 kilometers is Rs 118.

Why It Matters

Understanding APs helps in fields like AI/ML to analyze data patterns, in Physics to study motion with constant acceleration, and in Engineering to design structures with repeating elements. It's crucial for careers from data scientists to civil engineers and even financial analysts.

Common Mistakes

MISTAKE: Confusing the number of terms (n) with the value of the nth term (a_n) | CORRECTION: Remember 'n' is the position (like 5th day), while 'a_n' is the actual value at that position (like Rs 70 on the 5th day).

MISTAKE: Incorrectly identifying the first term (a) or common difference (d) from the word problem | CORRECTION: Read carefully! The first term is usually the starting value. The common difference is the constant increase or decrease per step.

MISTAKE: Using the formula for the nth term (a_n) when the sum of terms (S_n) is required, or vice versa | CORRECTION: Determine if the question asks for a specific item's value or the total accumulated value. 'How much did he save on the 10th day?' needs a_n. 'How much did he save in total over 10 days?' needs S_n.

Practice Questions

Try It Yourself

QUESTION: The cost of a mobile data pack is Rs 199 for the first month. For each subsequent month, the cost increases by Rs 10. What will be the cost of the data pack in the 5th month? | ANSWER: Rs 239

QUESTION: A gardener plants 15 saplings on the first day, 18 on the second, 21 on the third, and so on. If he continues this pattern, how many saplings will he plant in total over 7 days? | ANSWER: 175 saplings

QUESTION: A small business earns Rs 5000 in its first year. If its earnings increase by a fixed amount each year, and its earnings in the 4th year are Rs 6500, what will be its total earnings in the first 6 years? | ANSWER: Rs 36000

MCQ

Quick Quiz

A stadium has 20 seats in the first row, 24 in the second, 28 in the third, and so on. How many seats are there in the 10th row?

The Correct Answer Is:

The first term (a) is 20, and the common difference (d) is 4. Using the formula a_n = a + (n-1)d, for the 10th row (n=10), a_10 = 20 + (10-1)4 = 20 + 9*4 = 20 + 36 = 56. Oops, let's recheck. The options are tricky. The correct answer should be 20 + (10-1)*4 = 20 + 36 = 56. Let's correct my MCQ options. The correct option should be 56.

Real World Connection

In the Real World

Many financial schemes, like Recurring Deposits (RDs) or loan repayment schedules, can sometimes involve arithmetic progression patterns. For instance, calculating the total interest paid over time or the remaining balance might use AP concepts. Even planning for construction projects or managing inventory can involve APs to predict growth or depletion rates.

Key Vocabulary

Key Terms

ARITHMETIC PROGRESSION (AP): A sequence of numbers where the difference between consecutive terms is constant | FIRST TERM (a): The starting number in an AP | COMMON DIFFERENCE (d): The constant amount by which each term increases or decreases | Nth TERM (a_n): The value of the term at a specific position 'n' in the sequence | SUM OF N TERMS (S_n): The total value obtained by adding the first 'n' terms of an AP

What's Next

What to Learn Next

Great job mastering AP word problems! Next, you should explore Geometric Progressions (GP) and their word problems. GPs are similar but involve a common ratio instead of a common difference, opening up new ways to model growth and decay in real-world scenarios.