Statue of James Clerk Maxwell - Edinburgh Scotland. Photo by Ian Felgate 2014.
NUMBER PART 3
21. Direct and inverse relationships with linear proportions
Russian Fudge Recipe
Ingredients
3 cups White Sugar
120g butter
3 Tbsp. Golden Syrup
180ml sweetened condensed milk (half a standard tin)
Makes 18 pieces
Ingredients
3 cups White Sugar
120g butter
3 Tbsp. Golden Syrup
180ml sweetened condensed milk (half a standard tin)
Makes 18 pieces
The object here is to adjust the recipe so that it makes the required pieces in the table. The original recipe makes 18 pieces of fudge. 36 pieces is double the amount so just double the ingredients. 9 pieces is half the amount so halve the ingredients. To work out 45 pieces divide 45 by 18 = 45/18 = 5/2 = 2.5, so times all the ingredients by 2.5. To work out 6 pieces, divide 6 by 18 = 6/18 = 1/3, so times all the ingredients by 1/3 (or divide by 3).
But there is an easier way to work out the last two lines! Look at the column 'PIECES' and notice that 36 + 9 = 45 then the sugar column
6 + 1.5 = 7.5
Now continue this line of logic across the rest of the table until the third row is completed.
6 + 1.5 = 7.5
Now continue this line of logic across the rest of the table until the third row is completed.
Passion fruit Marshmallows
180ml passion fruit juice
6 tsps. Powdered gelatine
3 cups Caster Sugar
200ml water
3 egg whites
Makes 30 pieces
180ml passion fruit juice
6 tsps. Powdered gelatine
3 cups Caster Sugar
200ml water
3 egg whites
Makes 30 pieces
22. RATES AND RATIO
Ratios are expressions that compare quantities.
They are usually written using the double dot or colon symbol ':'
Eg 1. On a school camping trip the ratio of students to teachers was 3:1
this could also be represented as 3/1 or stated "three to one" but the colon symbol is most often used.
This means that for every 3 students there is 1 teacher. From this we can see that if there were 6 students there would be 2 teachers, (doubling the amounts equally in proportion), or 12 students and 4 teachers etc.
Notice that the ratio didn't tell us exactly 'how many' students or teachers were on the camp. But this will be used later to work this out given the total number of students and staff on the trip
Eg 2. On a life small lifestyle farm the ratio of horses to sheep to chickens was 2:3:4
This means that for every 2 horses on the farm there would be 3 sheep and 4 chickens.
The order of the subjects (horses, sheep and chickens) matches the ratio
2 horses
3 sheep
4 Chickens
They are usually written using the double dot or colon symbol ':'
Eg 1. On a school camping trip the ratio of students to teachers was 3:1
this could also be represented as 3/1 or stated "three to one" but the colon symbol is most often used.
This means that for every 3 students there is 1 teacher. From this we can see that if there were 6 students there would be 2 teachers, (doubling the amounts equally in proportion), or 12 students and 4 teachers etc.
Notice that the ratio didn't tell us exactly 'how many' students or teachers were on the camp. But this will be used later to work this out given the total number of students and staff on the trip
Eg 2. On a life small lifestyle farm the ratio of horses to sheep to chickens was 2:3:4
This means that for every 2 horses on the farm there would be 3 sheep and 4 chickens.
The order of the subjects (horses, sheep and chickens) matches the ratio
2 horses
3 sheep
4 Chickens
Examples
1) 20 students and teachers are on camp. The ratio of students to teachers is 3:1. How many students are on the camp?
First, form fractions from the ratio. If there are 3 students and 1 teacher that's 4 in total. So 3/4 are students and 1/4 are teachers. (In other words - add the ratios (3 +1 = 4) to get the common denominator.
Now multiply this fraction by the total number of people on the camp.
3/4x20=15 so there are 15 students on the camp, and……
1/4x20=5 teachers. (Or you could go 20 - 15 = 5 teachers).
1) 20 students and teachers are on camp. The ratio of students to teachers is 3:1. How many students are on the camp?
First, form fractions from the ratio. If there are 3 students and 1 teacher that's 4 in total. So 3/4 are students and 1/4 are teachers. (In other words - add the ratios (3 +1 = 4) to get the common denominator.
Now multiply this fraction by the total number of people on the camp.
3/4x20=15 so there are 15 students on the camp, and……
1/4x20=5 teachers. (Or you could go 20 - 15 = 5 teachers).
Further Examples
1)Bobby and Betty share $2000 in the ratio of 3:2
Calculate how much they each receive.
1) Make fractions. 3/5 and 2/5 (adding the ratio numbers together gives the denominator)
2) Bobby gets 3/5 of $2000 and Betty gets 2/5 of $2000
3/5 x 2000=$1200
Therefore, Betty must get $2000 - $1200= $800
2)Penny and Lani share $2400 in the ratio of 6:4
Calculate how much they each receive.
1) Make fractions. 6/10 and 4/10 (adding the ratio numbers together gives the denominator)
2) Penny gets 6/10 of $2400 and Lani gets 4/10 of $2400
3) Do the math for Penny
6/10 x 2400=$1440
Therefore, Lani must get $2400 - $1440
=$960.00
3)Nikau and Aria share $3200 in the ratio of 7:3
Calculate how much they each receive.
1) Make fractions. 7/10 and 3/10 (adding the ratio numbers together gives the denominator)
2) Nikau gets
7/10 x 3200=$2240
therefore Aria must get $3200 - $2240=$960.00
4)Janet and John win $500 with a school charity raffle ticket. They bought the ticket together, but Janet paid $4 and John paid $1. Calculate how much they each receive based on what they paid.
Janet gets 4/5 x $500=$400.00
Therefore, John gets 500-400=$100
5)Luke and Leia win $360 with a school charity raffle ticket. They bought the ticket together, but Luke $2 and Leia paid $4. Calculate how much they each receive based on what they paid.
Luke gets 2/6x360=$120
and Leia gets $360-120=$240.00
6)Mario and Serina win $640 with a school charity raffle ticket. They bought the ticket together, but Mario paid $3 and Serina paid $5. Calculate how much they each receive based on what they paid.
Mario gets 3/8x640=$240
and Serina gets $640-240=$400.00
Calculate how much they each receive.
1) Make fractions. 3/5 and 2/5 (adding the ratio numbers together gives the denominator)
2) Bobby gets 3/5 of $2000 and Betty gets 2/5 of $2000
3/5 x 2000=$1200
Therefore, Betty must get $2000 - $1200= $800
2)Penny and Lani share $2400 in the ratio of 6:4
Calculate how much they each receive.
1) Make fractions. 6/10 and 4/10 (adding the ratio numbers together gives the denominator)
2) Penny gets 6/10 of $2400 and Lani gets 4/10 of $2400
3) Do the math for Penny
6/10 x 2400=$1440
Therefore, Lani must get $2400 - $1440
=$960.00
3)Nikau and Aria share $3200 in the ratio of 7:3
Calculate how much they each receive.
1) Make fractions. 7/10 and 3/10 (adding the ratio numbers together gives the denominator)
2) Nikau gets
7/10 x 3200=$2240
therefore Aria must get $3200 - $2240=$960.00
4)Janet and John win $500 with a school charity raffle ticket. They bought the ticket together, but Janet paid $4 and John paid $1. Calculate how much they each receive based on what they paid.
Janet gets 4/5 x $500=$400.00
Therefore, John gets 500-400=$100
5)Luke and Leia win $360 with a school charity raffle ticket. They bought the ticket together, but Luke $2 and Leia paid $4. Calculate how much they each receive based on what they paid.
Luke gets 2/6x360=$120
and Leia gets $360-120=$240.00
6)Mario and Serina win $640 with a school charity raffle ticket. They bought the ticket together, but Mario paid $3 and Serina paid $5. Calculate how much they each receive based on what they paid.
Mario gets 3/8x640=$240
and Serina gets $640-240=$400.00
wages and Travelling Allowance
First, a note on the travel allowance. The travel allowance is paid even if you have to come into work for 1 hour. overtime pay. When an employee works more than their normal hours then they need to be paid for the overtime that they do. Some companies pay double time where you earn twice the normal rate per hour after you have worked a normal week. So if you earned say $16 per hour during your 40 hour week, any hours you work after that would earn $32 per hour. That is
$16 x 2 = $32,
Other companies might pay time and a half which is less than double time. Time and a half is where you earn your normal hourly rate PLUS half of your normal rate added on. So if you earned $16 per hour during the week, any hours you work after that would earn $24 per hour. That is
$16 + (1/2 x $16) = $24
A simpler method is to multiply your normal rate by (time and a half) or (1 + 1/2) = 1.5
1.5 x $16 = $24
The mistake some young employees make is thinking their overtime rate covers their 40 hour week. It only applies to the hours you work overtime. According to Employment New Zealand, companies don't have a legal requirement to pay overtime above the normal rate of pay after working a certain number of hours in a day or a week.
$16 x 2 = $32,
Other companies might pay time and a half which is less than double time. Time and a half is where you earn your normal hourly rate PLUS half of your normal rate added on. So if you earned $16 per hour during the week, any hours you work after that would earn $24 per hour. That is
$16 + (1/2 x $16) = $24
A simpler method is to multiply your normal rate by (time and a half) or (1 + 1/2) = 1.5
1.5 x $16 = $24
The mistake some young employees make is thinking their overtime rate covers their 40 hour week. It only applies to the hours you work overtime. According to Employment New Zealand, companies don't have a legal requirement to pay overtime above the normal rate of pay after working a certain number of hours in a day or a week.
A note on tax
The tax calculations used in these examples don't always reflect reality as they are simply a mathematical exercise. The objective is for students to be able to calculate a percentage of a quantity which has been shown earlier on the website.
The 10% rule for working out the tax is one worth practising when you don't have a calculator. So in Jamie's example below using the 10% rule
10% x 880 = 88
$88 x 2=$176.00
The 10% rule for working out the tax is one worth practising when you don't have a calculator. So in Jamie's example below using the 10% rule
10% x 880 = 88
$88 x 2=$176.00
1)Jamie works at Just Jeans earning $16 per hour plus time and a half overtime. (Overtime hours are those hours worked beyond 40 hours in one week). One week she worked 50 hours. (10 hours overtime).
a) How much did Jamie earn for the first 40 hours she worked?
16x40=$640
b) How much did Jamie earn for the last 10 hours at the overtime rate?
(16+8)x10=$240 or 1.5x16x10 = $240
c) How much did Jamie earn for the week (includes overtime)?
$640+$240=$880
d) Jamie pays tax on her income (including overtime). If she had to pay 20% tax, how much tax did she have to pay?
20/100 x $880=$176
a) How much did Jamie earn for the first 40 hours she worked?
16x40=$640
b) How much did Jamie earn for the last 10 hours at the overtime rate?
(16+8)x10=$240 or 1.5x16x10 = $240
c) How much did Jamie earn for the week (includes overtime)?
$640+$240=$880
d) Jamie pays tax on her income (including overtime). If she had to pay 20% tax, how much tax did she have to pay?
20/100 x $880=$176
2)Brad works at Beaurepaires earning $20 per hour plus time and a half overtime. (Overtime hours are those hours worked beyond 40 hours in one week). One week he worked 46 hours. (6 hours overtime).
a) How much did Brad earn for the first 40 hours he worked?
20x40=$800
b) How much did Brad earn for the last 6 hours at the overtime rate?
(20+10)x6=$180 or 1.5x$20x6=$180
c) How much did Brad earn for the week (including overtime)?
$800+$180=$980
d) Brad pays tax on his income (including overtime). If he had to pay 30% tax, how much tax did he have to pay?
30/100 x $980=$294 or the 10% rule
10% x $980=$98
$98 x 3=$294.00
a) How much did Brad earn for the first 40 hours he worked?
20x40=$800
b) How much did Brad earn for the last 6 hours at the overtime rate?
(20+10)x6=$180 or 1.5x$20x6=$180
c) How much did Brad earn for the week (including overtime)?
$800+$180=$980
d) Brad pays tax on his income (including overtime). If he had to pay 30% tax, how much tax did he have to pay?
30/100 x $980=$294 or the 10% rule
10% x $980=$98
$98 x 3=$294.00
3)Shane works at Star Bucks earning $22 per hour plus time and a half overtime. (Overtime hours are those hours worked beyond 40 hours in one week). One week he worked 44 hours.
a) How much did Shane earn for the first 40 hours he worked?
$22x40=$880
b) How much did Shane earn for the last 4 hours at the overtime rate?
(22+11)x4=$132 or 1.5x$22x4=$132
c) How much did Shane earn for the week (including overtime)?
$880+$132=$1,012
d) Shane pays tax on his income (including overtime). If he had to pay 24% tax, how much tax did he have to pay?
24/100 x $1012=$242.88
or the 10% rule
10% x $1012=$101.2
2 x $101.2 =$202.4 and for the remaining 4%, use the 1% rule and times it by 4
1% x $1012=$10.12
4 x $10.12=$40.48 therefore $202.4 + $40.48=$242.88
a) How much did Shane earn for the first 40 hours he worked?
$22x40=$880
b) How much did Shane earn for the last 4 hours at the overtime rate?
(22+11)x4=$132 or 1.5x$22x4=$132
c) How much did Shane earn for the week (including overtime)?
$880+$132=$1,012
d) Shane pays tax on his income (including overtime). If he had to pay 24% tax, how much tax did he have to pay?
24/100 x $1012=$242.88
or the 10% rule
10% x $1012=$101.2
2 x $101.2 =$202.4 and for the remaining 4%, use the 1% rule and times it by 4
1% x $1012=$10.12
4 x $10.12=$40.48 therefore $202.4 + $40.48=$242.88
4)Lance, an experienced forestry worker earns $26 per hour plus time and a half overtime. (Overtime hours are those hours worked beyond 40 hours in one week). One week he worked 48 hours.
a) How much did Lance earn for the first 40 hours he worked?
$26x40=$1,040
b) How much did Lance earn for the last 8 hours at the overtime rate?
(26+13)x8=$312
c) How much did Lance earn for the week (including overtime)?
$1040+$312=$1352
d) Lance pays tax on his income (including overtime). If he had to pay 30% tax, how much tax did he have to pay?
30/100 x $1352=$405.60
a) How much did Lance earn for the first 40 hours he worked?
$26x40=$1,040
b) How much did Lance earn for the last 8 hours at the overtime rate?
(26+13)x8=$312
c) How much did Lance earn for the week (including overtime)?
$1040+$312=$1352
d) Lance pays tax on his income (including overtime). If he had to pay 30% tax, how much tax did he have to pay?
30/100 x $1352=$405.60
5)Lucy works as a legal advisor earning $40 per hour plus time and a half overtime. (Overtime hours are those hours worked beyond 40 hours in one week). One week she worked 42 hours.
a) How much did she earn for the first 40 hours he worked?
$40x40=$1,600
b) How much did she earn for the last 2 hours at the overtime rate?
(40+20)x2=$120
c) How much did she earn for the week (including overtime)?
$1600+$120=$1720
d) She pays tax on her income (including overtime). If she had to pay 22% tax, how much tax did she have to pay?
22/100 x $1720=$378.40
a) How much did she earn for the first 40 hours he worked?
$40x40=$1,600
b) How much did she earn for the last 2 hours at the overtime rate?
(40+20)x2=$120
c) How much did she earn for the week (including overtime)?
$1600+$120=$1720
d) She pays tax on her income (including overtime). If she had to pay 22% tax, how much tax did she have to pay?
22/100 x $1720=$378.40
23. Find optimal solutions using numeric approaches - l6
prime factor trees
Earlier we learned that prime numbers are also used in prime factorisation. Factorisation means finding the factors of a number, except that they are prime factors. Numbers can be expressed as the product of prime numbers. For example, the number 240 below can be expressed as the product of its prime factors.
Find the prime factors of 240 and 320 using a prime factor tree. There is more than one way of listing the factors
Find the prime factors of 240 and 320 using a prime factor tree. There is more than one way of listing the factors
So the prime factors of 240 are: 2x2x2x2x3x5. When these are multiplied out we get 240.
Like wise the factors of 320 are: 2x2x2x2x2x2x5 = 320
However it is usual to write out the factors in exponential form as in below.
Like wise the factors of 320 are: 2x2x2x2x2x2x5 = 320
However it is usual to write out the factors in exponential form as in below.
Looking again at the first example, the prime factors of 240 could be achieved through many other initial factors.
Here the leading factors this time are 30 and 8. We could have also used 80 and 3. The follow on factors from there lead to the same prime factors as in the previous example.
use the prime factors in common to find the highest common factor (HCF) of two numbers
Using 240 and 320 as our example, we can find the highest common factor between them by comparing the prime factors or pairs they have in common and then multiplying them together.
240 = 2 x 2 x 2 x 2 x 3 x 5
320 = 2 x 2 x 2 x 2 x 2 x 2 x 5
There are 4 pairs of '2', and a 5 in common with both numbers. 3 is not a factor of 320 so it doesn't count. Now multiply the factors.
2 x 2 x 2 x 2 x 5 = 80 which is the highest common factor of 240 and 320.
240 = 2 x 2 x 2 x 2 x 3 x 5
320 = 2 x 2 x 2 x 2 x 2 x 2 x 5
There are 4 pairs of '2', and a 5 in common with both numbers. 3 is not a factor of 320 so it doesn't count. Now multiply the factors.
2 x 2 x 2 x 2 x 5 = 80 which is the highest common factor of 240 and 320.
examples
Sketch prime factor trees of the following numbers to determine their prime factors
1) 420
2) 250
3) 156
4) 600
5) 350
6) 180
7) 230
8) 380
9) 196
10 316
1) 420
2) 250
3) 156
4) 600
5) 350
6) 180
7) 230
8) 380
9) 196
10 316