In this question, letters of the alphabet are assigned number values:
A = 1, B = 2, C = 3, D = 4, E = 5, …
The product value of a word is created by multiplying the value of the letters e.g. the product value of ACE = 1 x 3 x 5 = 15
(a) What is the product value of the word PER?
(b) List the following words in order of their product value, starting with the lowest:
TAX TOIL HOLD ISLE
(c) Which three letter English word can be added to the word below so that the sum of their product values is equal to 413?
BAN + ……..
Write out the letters of the alphabet and assign numbers to the letters.
The product value of PER is 16 x 5 x 18
One way of working this out quickly is to say that the product value of PER is 18 lots of 16 x 5 and that 18 lots of 16 x 5 is 20 lots of 16 x 5 minus 2 lots of 16 x 5.
20 lots of 16 x 5 is 20 x 16 x 5 = 20 x 5 x 16 = 100 x 16 = 1600
2 lots of 16 x 5 is 2 x 16 x 5 = 2 x 5 x 16 = 10 x 16 = 160
so 18 lots of 16 x 5 is 1600 – 160 = 1440
(b) TAX, HOLD, ISLE, TOIL
TAX = 20 x 1 x 24 = 20 x 24
TOIL = 20 x 15 x 9 x 12 = 108 x 15 x 20
HOLD = 8 x 15 x 12 x 4 = 60 x 8 x 12
ISLE = 9 x 19 x 12 x 5 = 60 x 9 x 19
There is no need to calculate all of these products.
We can see that the product value of TOIL is bigger than the product value of ISLE and that the product value of ISLE is bigger than the product value of HOLD and the product value of HOLD is bigger than the product value of TAX
So in order from lowest to highest, the words are TAX, HOLD, ISLE, TOIL
The product value of the word BAN is 2 x 1 x 14 = 28
So we need to find a three letter English word whose product value is 413 – 28 i.e. whose product value is 385.
To do this we need to find 3 factors of 385 which when multiplied together produce a value of 385 and we need to see if the letter values for these three factors make an English three letter word.
In this type of question, it is highly likely that the 3 factors will be primes because that will cut down the time needed to work out the solution. However, that is not necessarily the case, and if the product is not a product of just 3 primes, then you will need to consider other non-prime factors such as the number 1, which represents the letter A, and which is a factor of every number. However, in general, try using the method of prime factorisation first before you consider other possibilities.
To find the factors, we can start by using the method of prime factorisation.
Doing this, we have that 385 = 5 x 77 = 5 x 7 x 11
The letter values for these factors are E G and K and they form the word KEG which has a product value of 11 x 5 x 7 = 385