UNCORRECTEDSAMPLEPAGES

The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.

Prime number landscaping

1 The team at Prime Landscaping only likes to plant trees, lay sleepers, or group large rocks in clumps where the number of items in the group is a prime number. The landscaping team are interested in the prime number options available to them and the total number of objects that they might need to purchase for particular jobs.

a What size groupings can Prime Landscaping arrange a pile of large rocks that have less than 15 in one group?

b Would Prime Landscaping ever lay a group of 27 sleepers? Give a reason for your answer.

One of Prime Landscaping’s clients said they only wanted trees or shrubs that were planted in even numbers as they disliked odd numbers of plants (such as a group of 3 trees or a group of 7 shrubs). Prime Landscaping felt confident they could meet the needs of their client and still stay true to their policy of planting only in groups of prime numbers.

c What size group of plants did Prime Landscaping plant for their client?

A new client purchased 59 screening plants and wanted them to be planted in three different, but similar sized groups. Prime Landscaping said that this was possible and proceeded to plant the plants.

d What size groups were the screening plants planted in?

Blinking rates

2 Diviesh tends to blink his eyes every four seconds, while Jordan blinks his eyes only every seven seconds. Blinking provides moisture to the eyes and prevents irritation and damage. Diviesh and Jordan are interested in the number of times they will blink their eyes during various time intervals and how often they will blink at the same time depending on their blinking rates.

a If Diviesh and Jordan both blink together, how long will it be until they blink together again?

b How many times will Diviesh and Jordan blink together in one minute, and at what times will this occur?

Concentrated staring at a computer monitor can significantly reduce the eye blinking rate, and can result in what is known as ‘gamer eye syndrome’.

Diviesh and Jordan were filmed playing a one-hour gaming session. Diviesh’s blinking rate reduced to 3 blinks per minute, and Jordan’s blinking rate reduced to 4 blinks per minute.

c At the start of a particular three-minute time period in their computer gaming session, Diviesh and Jordan both blinked together. If this time was referred to as 0 seconds, write down the time (in seconds) of Diviesh’s next blinks and Jordan’s next 10 blinks.

d How frequently do Diviesh and Jordan blink together during a one-hour computer gaming session?

Excessive blinking can indicate a disorder of the nervous system. A doctor examined a particular patient and initially observed that their blink rate was 30 blinks per minute. She then decided to monitor the patient for a period of 5 minutes and was relieved to observe the patient only blinked 110 times.

e What was the average blinking rate (blinks per minute) for the patient over the 5 minute period?

A reluctant reader

3 A keen student mathematician but reluctant reader, was encouraged by his Maths teacher to read just two pages at night on the first night and then simply add another two pages each night – so on the second night the student would have to read four pages, and then six pages on the third night and so on. The student agreed to do this.

The student is interested in how many pages he will be reading on particular nights and how long it will take to read a book of a certain size.

a How many pages would the student be reading on the 5th night?

b On what night was the student reading 16 pages?

c How many nights did it take the student to read a 182 page book?

d How big a book can the student read in a 30-day month?

e If a student wishes to read a 275 page book in 10 days, describe a different rule with a different pattern that would allow this to occur. List the number of pages the student would need to read each night.