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Multiplying and dividing decimal numbers by 10; 100 and 1,000
Example 1
The most skilled goldsmiths can make gold plates as thin as \latex{ 0.0015 } \latex{ mm }.
a) How thick would \latex{ 10 } gold plates be if they were stacked on top of each other?
b) Would \latex{ 1,000 } stacked gold plates have a thickness of at least \latex{ 1 } \latex{ mm ?}
b) Would \latex{ 1,000 } stacked gold plates have a thickness of at least \latex{ 1 } \latex{ mm ?}
Solution
a) \latex{0.0015\times10=\frac{15}{10000}\times10=\frac{15}{1000}=0.015.}
The stack of ten gold plates would be \latex{ 0.015 } \latex{ mm } thick.
b) \latex{0.0015\times1000=\frac{15}{10000}\times1000=\frac{15}{10}=1.5.}
The thickness of a thousand gold plates would be \latex{ 1.5 } \latex{ mm }, which is more than \latex{ 1 } \latex{ mm }.
Examine how multiplication by \latex{ 10; 100 } and \latex{ 1,000 } works with the help of the place value table.
\latex{100s}
\latex{10s}
\latex{1s}
\latex{10th}
\latex{100th}
\latex{1000th}
\latex{10000th}
\latex{ 5 }
\latex{ 1 }
\latex{ 5 }
\latex{ 1 }
\latex{ 5 }
\latex{ 1 }
\latex{ 5 }
\latex{ 1 }
\latex{ 5 }
\latex{ 1 }
\latex{ 0 }
\latex{ 5 }
\latex{ 1 }
\latex{0.0015}
\latex{0.0015 \times 1\textcolor{#E3004F}{0}=0.015}
\latex{0.0015 \times 1\textcolor{#E3004F}{00}=0.15}
\latex{0.0015 \times 1,\textcolor{#E3004F}{000}=1.5}
\latex{0.0015 \times 1\textcolor{#E3004F}{0,000}=15}
\latex{0.0015 \times 1\textcolor{#E3004F}{00,000}=150}
When you multiply a decimal number by \latex{ 10; 100 } or \latex{ 1,000 }, each digit of the decimal moves to a place value that is greater by one, two or three. Additional zeros in the multiplier result in further place value movements.
Dividing decimals by 10; 100 and 1,000
Example 2
- If you put \latex{ 10 } €\latex{ 1 } coins in a stack, the stack is \latex{ 23.3 } \latex{ mm } high. How thick is a single €\latex{ 1 } coin?
- How thick is a €\latex{ 100 } banknote if a stack of \latex{ 100 } banknotes is \latex{ 1.2 } \latex{ cm } thick?
Solution
a) \latex{23.3\div10=\frac{233}{10}\div10=\frac{233}{100}=2.33}
The thickness of a €\latex{ 1 } coin is \latex{ 2.33 } \latex{ millimetres }.
b) \latex{1.2\div100=\frac{12}{10}\div100=\frac{12}{1000}=0.012}.
The thickness of a €\latex{ 100 } banknote is \latex{ 0.012 } \latex{ cm = 0.12 } \latex{ mm }.
Examine how division by \latex{ 10; 100 } and \latex{ 1,000 } works with the help of the place value table.
\latex{10s}
\latex{1s}
\latex{10th}
\latex{100th}
\latex{1000th}
\latex{10000th}
\latex{100000th}
\latex{ 5 }
\latex{ 1 }
\latex{ 5 }
\latex{ 1 }
\latex{ 5 }
\latex{ 3 }
\latex{ 5 }
\latex{ 1 }
\latex{ 5 }
\latex{ 1 }
\latex{13.5}
\latex{13.5\div 1\textcolor{#E3004F}{0}=1.35}
\latex{13.5\div 1\textcolor{#E3004F}{00}=0.135}
\latex{13.5 \div1,\textcolor{#E3004F}{000}=0.0135}
\latex{13.5\div 1\textcolor{#E3004F}{0,000}=0.00135}
\latex{ 3 }
\latex{ 3 }
\latex{ 1 }
\latex{ 3 }
\latex{ 3 }
When you divide a decimal by \latex{ 10; 100 } or \latex{ 1,000 }, each digit of the decimal number moves to a place value that is lesser by one, two or three. Additional zeros in the divisor result in further place value movements.
Exercises
{{exercise_number}}. Multiply the following numbers by ten, one hundred and one thousand.
a) \latex{ 152 }
b) \latex{ 0.3 }
c) \latex{ 0.8 }
d) \latex{ 0.12 }
e) \latex{ 0.072 }
f) \latex{ 0.152 }
g) \latex{ 0.06 }
h) \latex{ 0.0012 }
i) \latex{ 0.0515 }
f) \latex{ 0.0102 }
k) \latex{ 0.5003 }
l) \latex{ 1.03 }
m) \latex{ 10.04 }
n) \latex{ 22.08 }
o) \latex{ 4.017 }
p) \latex{ 70.202 }
{{exercise_number}}. What are the tenths, hundredths and thousandths of the following numbers?
a) \latex{ 15.2 }
b) \latex{ 30 }
c) \latex{ 0.8 }
d) \latex{ 0.12 }
e) \latex{ 0.72 }
f) \latex{ 1.52 }
g) \latex{ 0.06 }
h) \latex{ 0.012 }
i) \latex{ 0.515 }
j) \latex{ 0.102 }
k) \latex{ 0.503 }
l) \latex{ 1.03 }
m) \latex{ 10.4 }
n) \latex{ 22.08 }
o) \latex{ 4.017 }
p) \latex{ 70.202 }
{{exercise_number}}. Continue the sequence of numbers by multiplying each member by ten. Write three new members for each sequence.
a) \latex{ 0.0001; \;0.001; ... }
b) \latex{ 0.0023; \;0.023; ... }
c) \latex{ 1.0005; \;10.005; ... }
d) \latex{ 0.76543; \; 7.6543; ... }
{{exercise_number}}. Continue the sequence of numbers by dividing each member by ten. Write three new members for each sequence.
a) \latex{ 0.1; \;0.01; ... }
b) \latex{ 23; \;2.3; ... }
c) \latex{ 1.05; \;0.105; ... }
d) \latex{ 7654.3; \;765.43; ... }
{{exercise_number}}. Convert the following quantities into \latex{ centimetres }.
a) \latex{ 2.3 } \latex{ m }
b) \latex{ 4.7 } \latex{ m }
c) \latex{ 13.27 } \latex{ m }
d) \latex{ 3.082 } \latex{ km }
e) \latex{ 42 } \latex{ mm }
f) \latex{ 0.7 } \latex{ m }
g) \latex{ 123 } \latex{ mm }
h) \latex{ 1.2 } \latex{ km }
{{exercise_number}}. Convert the following quantities into \latex{ decimetres }.
a) \latex{ 3.75 } \latex{ m }
b) \latex{ 4.7 } \latex{ cm }
c) \latex{ 38.5 } \latex{ mm }
d) \latex{ 0.3 } \latex{ km }
e) \latex{ 0.85 } \latex{ mm }
f) \latex{ 32.7 } \latex{ cm }
g) \latex{ 4.5 } \latex{ mm }
h) \latex{ 0.05 } \latex{ km }
{{exercise_number}}. Convert the following quantities into \latex{ kilograms }.
a) \latex{ 1.2 } \latex{ t }
b) \latex{ 0.03 } \latex{ t }
c) \latex{ 4,570 } \latex{ g }
d) \latex{ 380 } \latex{ g }
e) \latex{ 0.8 } \latex{ t }
f) \latex{ 178 } \latex{ g }
g) \latex{ 120 } \latex{ g }
h) \latex{ 34.5 } \latex{ g }
{{exercise_number}}. On average, a car uses \latex{ 6.2 } \latex{ litre }s of petrol every \latex{ 100 } \latex{ kilometres }. Assuming a constant consumption rate, how much petrol would it take to drive the following distances?
a) \latex{ 1,000 } \latex{ km }
b) \latex{ 10 } \latex{ km }
c) \latex{ 1 } \latex{ km }
d) \latex{ 100 } \latex{ m }
{{exercise_number}}. How thick is a banknote, if \latex{ 1,000 } notes are \latex{ 9 } \latex{ cm } thick?
{{exercise_number}}. A sheet of paper is \latex{ 0.11 } \latex{ mm } thick. One photocopy machine has a \latex{ 100 }-sheet paper tray and another one has a \latex{ 1,000 }-sheet paper tray. What total thickness of paper can fit in each machine?
{{exercise_number}}. Express the same quantities in smaller units.
a) \latex{ 0.056 } \latex{ km }
b) \latex{ 0.000 56 } \latex{ km^{2} }
c) \latex{ 0.000 0056 } \latex{ m^{3} }
{{exercise_number}}. Express the same quantities in larger units.
a) \latex{ 73,000 } \latex{ mm }
b) \latex{ 73, 000, 000 } \latex{ mm^{2} }
c) \latex{ 0.056 } \latex{ m^{3} }
{{exercise_number}}. The world's smallest marine fish has been discovered on the Great Barrier Reef. The stout infantfish can grow to a length of \latex{ 7 } to \latex{ 8 } millimetres and has a mass of \latex{ 0.001 } \latex{ grams }. How many grams do \latex{ 100 } infantfish weigh?
{{exercise_number}}. The weight of one parasitic wasp is \latex{ 0.000005 } \latex{ gram }s. How many \latex{ grams } do \latex{ 10; } \latex{ 100; } \latex{ 1,000; } \latex{ 10, 000 } and \latex{ 100, 000 } wasps weigh?
{{exercise_number}}. A dwarf mosquito flaps its wings \latex{ 62,760 } times a \latex{ minute. } One single mosquito wing contraction takes \latex{ 0.00045 } \latex{ seconds }. How long do one million contractions take?
Quiz
What is the product of the following multiplication?
\latex{100,000 \times 10,000 \times 1,000 \times 100 \times 10 \times 0.1 \times 0.01 \times 0.001 \times 0.0001 \times 0.00001}
\latex{100,000 \times 10,000 \times 1,000 \times 100 \times 10 \times 0.1 \times 0.01 \times 0.001 \times 0.0001 \times 0.00001}
