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Multiplying and dividing decimal numbers by natural numbers
Example 1
The thickness of a €\latex{ 2 } coin is \latex{ 2.2 } \latex{ millimetres }. How high would a stack of \latex{ 24 } coins be?
Solution 1
(you can convert and multiply the amount as a multiplication of mixed numbers)
The thickness of \latex{ 24 } coins in mm:
\latex{2.2\times24=2\frac{2}{10}\times24=\frac{22}{10}\times24=\frac{528}{10}=52.8}
Solution 2
multiplication in writing
\latex{ 2\;. }
\latex{ 2 }
\latex{ \times }
\latex{ 2 }
\latex{ 4 }
\latex{ 4 }
\latex{ 4 }
\latex{ 8 }
\latex{ 8 }
\latex{ 5 }
\latex{ 2\;. }
\latex{ 8 }
Do not put any decimal points in the partial product.
If you stack \latex{ 24 } €\latex{ 2 } coins on top of each other, the stack will be \latex{ 52.8 } \latex{ mm } high.
If you stack \latex{ 24 } €\latex{ 2 } coins on top of each other, the stack will be \latex{ 52.8 } \latex{ mm } high.
\latex{22\times24=528}
\latex{2.2\times24=52.8}
÷10
÷10
When multiplying a decimal number by a natural number, treat the decimal as if it were a whole number, except that you must use as many fractions in the product as there are fractions in the original number.
\latex{6\times4=24}
\latex{0.6\times4=2.4}
\latex{0.06\times4=0.24}
\latex{0.006\times4=0.024}
Any decimal number multiplied by \latex{ 0 } is \latex{ 0 }.
Dividing decimal numbers by natural numbers
Example 2
Three gold miners have found \latex{ 46.2 } \latex{ dkg } of gold. How many \latex{ dkg } of gold do they each get if they share it equally?
Solution 1
\latex{ 46.2 } \latex{ dkg = 462 } \latex{ g }, each gold digger gets \latex{ 462 } \latex{ g } \latex{ \div 3=154 } \latex{ g } of gold.
Solution 2
(division in writing)
\latex{ 4 }
\latex{ 6. }
\latex{ 2 }
\latex{ 3 }
\latex{ 1 }
\latex{ 1 }
\latex{ 6 }
\latex{ 1 }
\latex{ 2 }
\latex{ 0 }
\latex{ \div }
\latex{ = }
\latex{ 5. }
\latex{ 4 }
If there is a decimal point in the divident, include it in the quotient as well.
They each get \latex{ 154 } \latex{ g = 15.4 } \latex{ dkg } of gold.
\latex{462\div3=154}
\latex{46.2\div3=15.4}
÷10
÷10
When dividing a decimal number by a natural number, treat the decimal as if it were a whole number. However, when you reach the decimal point in the dividend, you must include it in the quotient.
\latex{24\div3=8}
\latex{2.4\div3=0.8}
\latex{0.24\div3=0.08}
\latex{0.024\div3=0.008}
Division by \latex{ 0 } is undefined in the case of decimal numbers.
Exercises
{{exercise_number}}. Perform the following multiplications in your head.
a) \latex{15\times4}
b) \latex{18\times2}
c) \latex{12\times7}
d) \latex{25\times4}
e) \latex{125\times8}
\latex{1.5\times4}
\latex{1.8\times2}
\latex{1.2\times7}
\latex{2.5\times4}
\latex{12.5\times8}
\latex{0.15\times4}
\latex{0.18\times2}
\latex{0.12\times7}
\latex{0.25\times4}
\latex{1.25\times8}
{{exercise_number}}. Practice some more by performing the following multiplications.
a) \latex{57\times14}
b) \latex{78\times91}
c) \latex{13.5\times176}
d) \latex{97\times208}
\latex{5.7\times14}
\latex{7.8\times91}
\latex{1.35\times176}
\latex{9.7\times208}
\latex{0.57\times14}
\latex{0.78\times91}
\latex{0.135\times176}
\latex{0.97\times208}
\latex{0.057\times14}
\latex{0.078\times91}
\latex{0.0135\times176}
\latex{0.097\times208}
\latex{0.0057\times14}
\latex{0.0078\times91}
\latex{0.00135\times176}
\latex{0.0097\times208}
{{exercise_number}}. A single wood parquet tile is \latex{ 3.5 } \latex{ cm } wide. How wide are \latex{ 12 } of these tiles?
{{exercise_number}}. If a CD case is \latex{ 1.05 } \latex{ cm } thick, what is the height of a stack of \latex{ 15 } CD cases?
{{exercise_number}}. Perform the following multiplications
a) \latex{25.6\times142}
b) \latex{47.3\times93}
c) \latex{1.25\times80}
d) \latex{12.5\times107}
e) \latex{37.4\times559}
f) \latex{80.06\times147}
g) \latex{1.025\times444}
h) \latex{32.125\times320}
i) \latex{2,003.072\times225}
{{exercise_number}}. A coin collector wraps his old ¢\latex{ 1 }, ¢\latex{ 10 } and ¢\latex{ 50 } coins in rolls of \latex{ 50 }. How long will such a roll be if
a) a ¢\latex{ 1 } coin is \latex{ 0.167 } \latex{ cm } thick;
b) a ¢\latex{ 10 } coin is \latex{ 0.193 } \latex{ cm } thick;
c) a ¢\latex{ 50 } coin is \latex{ 0.238 } \latex{ cm } thick?
{{exercise_number}}. The collector is now packing his €\latex{ 1 }, €\latex{ 2 }, and his collectors' series €\latex{ 5 } coins in rolls of \latex{ 20 }. How long is a roll if
a) a €\latex{ 1 } coin is \latex{ 2.33 } \latex{ mm } thick;
b) a €\latex{ 2 } coin is \latex{ 2.2 } \latex{ cm } thick;
c) a €\latex{ 5 } coin is \latex{ 2.38 } \latex{ mm } thick?
{{exercise_number}}. If a book has \latex{ 408 } pages and a single page is \latex{ 0.12 } \latex{ mm } thick, how thick is the book without the covers?
{{exercise_number}}. Perform the following divisions in your head.
a) \latex{36\div4}
b) \latex{18\div2}
c) \latex{42\div7}
d) \latex{120\div5}
e) \latex{108\div9}
\latex{3.6\div4}
\latex{1.8\div2}
\latex{4.2\div7}
\latex{12\div5}
\latex{1.08\div9}
\latex{0.36\div4}
\latex{0.18\div2}
\latex{4.2\div7}
\latex{12\div5}
\latex{0.0108\div9}
{{exercise_number}}. Practice some more by performing the following divisions.
a) \latex{686\div7}
b) \latex{45.6\div12}
c) \latex{586.8\div18}
d) \latex{578\div17}
\latex{68.6\div7}
\latex{4.56\div12}
\latex{58.68\div18}
\latex{57.8\div17}
\latex{6.86\div7}
\latex{0.456\div12}
\latex{5.868\div18}
\latex{5.78\div17}
\latex{0.686\div7}
\latex{0.0456\div12}
\latex{0.5868\div18}
\latex{0.578\div17}
\latex{0.0686\div7}
\latex{0.00456\div12}
\latex{0.05868\div18}
\latex{0.0578\div17}
{{exercise_number}}. A \latex{ 648 }-page encyclopaedia is \latex{ 6.48 } \latex{ cm } thick without covers. How thick is one page of the book?
{{exercise_number}}.
a) What is the quotient if the divisor is \latex{ 24 } and the dividend is \latex{ 31.2 ?}
b) What is the divisor if the quotient is \latex{ 12 } and the dividend is \latex{ 408.96 ?}
c) What is the dividend if the divisor is \latex{ 47 } and the quotient is \latex{ 23.9 ?}
b) What is the divisor if the quotient is \latex{ 12 } and the dividend is \latex{ 408.96 ?}
c) What is the dividend if the divisor is \latex{ 47 } and the quotient is \latex{ 23.9 ?}
{{exercise_number}}. The next member in the sequence is obtained by multiplying the previous member by \latex{ 13 }. What numbers should be written instead of the letters?
a) \latex{ 0.7;} \latex{ 9.1; } \latex{ 118.3; } \latex{ A }; \latex{ B }; \latex{ C }.
b) \latex{ A }; \latex{ B }; \latex{ 16.9; } \latex{ 219.7; } \latex{ 2856.1; } \latex{ C }.
{{exercise_number}}. The next member in the sequence is obtained by dividing the previous member by \latex{ 12 }. What numbers should be written instead of the letters?
a) \latex{ 12441.6 }; \latex{ 1036.8 }; \latex{ A }; \latex{ B }; \latex{ C }; \latex{ D }.
b) \latex{ A }; \latex{ B }; \latex{ 1.728 }; \latex{ 0.144 }; \latex{ C }; \latex{ D }.
{{exercise_number}}. The following measures of length are commonly used in the UK: \latex{ 1 } \latex{ inch } \latex{ = 2.54 } \latex{ cm };
\latex{ 1 } \latex{ foot = 30.48 } \latex{ cm }; \latex{ 1 } \latex{ yard } \latex{ = 0.9144 } \latex{ m }; \latex{ 1 } \latex{ mile = 1.609 } \latex{ km }.
a) How many \latex{ centimetres } is the diagonal of a \latex{ 22 }-\latex{ inch } monitor screen?
b) What is the height in \latex{ centimetres } of a man who is exactly \latex{ 6 } \latex{ feet } tall?
c) Which distance is easier for a runner? \latex{ 100 } \latex{ metres } or \latex{ 110 } \latex{ yards }?
d) How many \latex{ kilometres } is the distance from London to Manchester if that distance is \latex{ 185 } \latex{ miles }?
{{exercise_number}}. The Smith household used \latex{ 110.642 } \latex{ m^{3} } of gas in \latex{ 1 } \latex{ week. } On average, how much gas did they use \latex{ per } \latex{ day? }
{{exercise_number}}. There are \latex{ 32 } cards in a deck that is \latex{ 1.2 } \latex{ cm } thick. How many \latex{ centimetres } thick is a single card?
{{exercise_number}}. If there are \latex{ 52 } cards in a deck, and you stack all the cards from two decks on top of each other, the resulting stack is \latex{ 3.12 } \latex{ cm } thick. How thick is a single card?
{{exercise_number}}. A Hungarian tourist bought \latex{ 265 } euros for his forthcoming trip to Austria. On that day, \latex{ 1 } euro was equal to \latex{ 400 } Hungarian forints (HUF). How many HUF did he get back in change if he paid with \latex{ 10, 000 } HUF banknotes?
Quiz
Steve did a multiplication on a piece of paper, but his little brother tore the paper up. Only a part of the multiplication is visible on the remaining scrap of paper. What whole number was the multiplier?
