Arith Series - Applied

1. Sarah has recently begun her driving lessons so that she can obtain her licence after her exams in November of this year.

Her first lesson was 20 minutes, her second lesson was 30 minutes and each subsequent lesson was 10 minutes longer than the previous lesson.

(i) How long will Sarah's 25th lesson last?

(ii) How many hours of driving instruction will Sarah have completed after her 25th lesson?

(iii) During which lesson will Sarah have completed 70 hours of instruction?

Hint.Be careful of the time units!!.
Answer.2(i) 25^th lesson: 4 hours 20 min.
(ii) Total - 58 hours 20 mins.
.(iii) During the 28th lesson.

2. David is saving to buy a new car. He is planning to spend $16,000 on the car.

He already has $7,500 saved but now plans to save more seriously by putting more money aside from his wages.

He begins by putting $100 aside in the first month. The next month he puts $150 aside and in the third month he puts $200 aside.

(i) If David continues to increase the amount he puts aside by $50 per month, how many months does it take him to reach his planned budget?

(ii) How much would David have to save in the last month of his plan?
Comment on that amount. Is there an alternative strategy for David?

Answer.(i) 17 more months.
(ii) Saving $900.

3. Ellen is being offered a Traineeship while she competes her University courses. She can elect to choose between two different salary packages.

Package 1: A starting salary of $30,000 and an annual increase of $2,250 per year.

Package 2: A starting salary of $26,500 and an annual increase of $3,500 per year.

(i) Show that, at the end of 5 years, Ellen's salary with Package 1 is $39,000.

(ii) Show that the total income Ellen receives over 5 years under Package 2
is $167,500.

(iii) Which package would be better for Ellen to select? Explain your reasons.

4. A very tall building with 106 floors is to be constructed.

The first floor will be built at a cost of $3 million. Each subsequent floor will cost $0.5 million more than the previous floor.

(i) What will be the cost of the 42nd floor?

(ii) What will be the cost of the building on this basis when it is completed (ignore cost overruns)?

Answer.(i) Floor 42 cost $23.5 million.
(ii) Total = $2,881.5 million.

5. Damien and Alyssa are employed by an accounting firm.

Damien is offered a salary of $600 per week with an increase of $50 per week at the beginning of each year.

Alyssa is offered a salary of $500 per week with an increase of $70 per week at the beginning of each year.

By developing two sequences for their respective salaries, find in which year of employment Alyssa will be earning more that Damien?

Hint.Be very careful how you interpret your answer.
Answer.(i) Floor 42 cost $23.5 million.
(ii) Total = $2,881.5 million.

6. In the lead up to the swimming World Championships, one of the Australian swimmers was placed on a new training regime by his coach on returning from injury. On the first day, he had to complete 26 laps of the 50 m pool.

On each succeeding day, the laps to be completed increased by 6 (hence to 32 laps, 38 laps, etc) until 200 laps were being completed. At that stage, the swimmer had to maintain the 200 laps per day for another 14 days.

(i) On which day, after returning from injury, did the swimmer first swim the 200 laps?

(ii) What was the total distance (in km) the swimmer completed by the end of this training schedule?

Answer.(i) Day 30.
(ii) Total = 309.5 km.

7. A timber worker is stacking logs. The logs are stacked in layers so that each layer has 2 less logs than are stacked in the row below.

There are 8 logs in the top row, 10 in the next row and so on. There are n rows altogether.

(i) Write down the number of logs in the bottom row.

(ii) Show that the number of logs stacked in the pile can be expressed as n² + 7n.

Answer.(i) No of logs = 6 + 2n.

8. During an examination, the supervisor stands 3 metres in front of the first desk in a column of desks holding a box of tissues, Students are seated in columns of desks with 1 metre from the front of one desk to the front of the next desk. There are 25 desks in a column.

All 25 students in a particular column of desks take turns in asking for a tissue. Each student waits until the supervisor returns to the supervising position 3 metres in front of the first desk of the column, before putting their hand up and requesting a tissue.

How many metres will the supervisor have to walk to provide each of the 25 students with one tissue and returning to the position 3 metres in front of the first desk?

Answer.Distance = 750 m.

9. Concrete blocks are used to build a wall to stop erosion. Each block is 1.5 m wide.

The lowest row (we will call it Row 1) is 180 m long. Each of the rows above the base row
(numbered 2 to 20) have 3 fewer blocks than the one below.

Above Row 20, each of the rows have 1 less block than the row below. That pattern continues until the top row which has a total of 10 blocks.

(i) How many blocks are in row 20?

(ii) What is the total number of rows in the wall?

(iii) How many blocks were used to construct this wall?

Answer.(i) 120 blocks.
(ii) 73 rows.
(iii) 3,738 blocks.

10. Susan is very fond of horses. She has a large paddock with a series of troughs into which she dumps a large bucket of hay for the horses. Susan walks from her truck, pours the hay into the trough and then walks back to her truck. She does that in turn for all the toughs in the paddock.

The first trough is 50 m from the gate where her truck is parked. The second trough is 50 m further and the other 8 troughs are also 50 m further apart.

(i) How far did Susan walk from her truck to the third trough and back?

(ii) How far did Susan walk from her truck to the last trough and back?

(iii) How far did Susan walk to fill up all ten troughs and return to her truck?

Answer.(i) 300 m.
(ii) 1,000 m.
(iii) Total distance = 5.5 km.