Increasing the basic rate
A. The answer given is D: Pounds (11.2 X 2.5100 + 11.2). This isn't how I teach my students; I would tell them to multiply 11.2 by 1.025. Does this mean that my way is not a good one to show them?
A learner benefits most by having a variety of methods for solving problems available to them. When I first read this question my immediate thought was that 21Z2% is 10% V 4.
10% of pound;11.20 is pound;1.12; 10% V 2 = 5%, so 5% (pound;1.12 V 2) = 56p; 5% V 2 = 21Z2 %, so 21Z2% (56p V 228p.
For an increase of 21Z2%, the electrician's basic rate will now be pound;11.20 + 28p (100% + 21Z2%pound;11.48.
The way you describe is also very quick; it shows that 0.025 is the amount the whole of the salary (1) is increased by. Hence 1 + 0.025 = 1.025 of the original amount of pound;11.20: 1.025 x pound;11.20 = pound;11.48.
The answer D changes 21Z2% into a fraction, hence 2.5100, then finds what 2.5100 of pound;11.20 is: 2.5100 x pound;11.20 ("of" is another way of saying "x"). This does, of course, give 0.025 x pound;11.20. Then this amount has to be added to the original: 2.5100 x pound;11.20 + pound;11.20 = Pounds 11.48. Learners aren't allowed calculators in their tests so going through each one to determine the correct answers is much more cumbersome. Another method that you may use recognises that: JJ x 10 2.5 J=JJ25JJ= 1 givingJpound;11.20 + pound;11.20 100JJJ1000JJ 40JJJJJJ 40 JJ x 10 All are very closely related. Exposing learners to different approaches to percentage problems increases understanding of the concept.
When I first started teaching, I really did not have a feel for the concept of percentage; to solve questions involving percentage I used to use algebra.
Some people say they would use a calculator anyway. Use of the percentage button can vary between calculators; for the Casio fx-83MS the process is as follows: type in 11.20 then "x" 2.5 (or whatever percentage you wish to find), followed by the "shift" button and the "%" button. The calculator responds with "0.28" (2.5% of the amount). We wish this to be added to pound;11.20, so now type in "+" (do not press "=" in between). The calculator adds the 0.28 to the original 11.20, and displays "11.48".
Some may think this was a really long-winded approach to a simple problem, and that all you need to understand is that pound;11.20 is our original amount and that this represents the one whole. The electrician's new earnings will be 1.025 greater than the original, so, type in 11.20 x 1.025 and "=" - much quicker.
The key to understanding the calculation is that the original is the 100% or the whole and we can add or subtract from this amount. This naturally then leads on to problems of the type where the "extra" or "less" is included in the amount you are given. For example, when you are given a price that includes VAT of 17.5%. The amount you have is 117.5% - the original 100% + VAT of 17.5%.
Q. What is the difference between the Golden Section and Divine Proportion?
A. They are the same: Golden Section (Greeks), Divine Proportion (Renaissance artists). An article in New Scientist (December 2128, 2002) about quasicrystals inspired this poem below. This number occurs naturally over and over again.
A Golden Ratio, Phi The Phi of ...nature arranging leaves efficiently.
...crustaceans creating spiral homes.
...Science creating vertical steps in quasicrystals.
...Art, the perfect rectangle for form of face.
...Architecture in the sculpture of stone.
...Astrophysics: rotating blackholes, changing specific heat.
...mathematics, division of consecutive terms of Fibonacci.
...value (1 + C5)2 or 1.6180339887I ...a single sunflower seed shaping its head.
...the universe, timeless and infinite.
...Luca Pacioli: a metaphor for God.
...the Golden Ratio, eternal perfection.
* Luca Pacioli was a 15th-century Italian Franciscan friar and mathematician who wrote a treatise called "The Divine Proportion"