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Changes of the product

€\latex{ 8 }

A school librarian wants to buy \latex{ 10 } copies of The Hungry Caterpillar. The book costs \latex{ 8 } euros today, but next week it will be half the price. How much would \latex{ 10 } books cost now? How much will they cost next week?

Now: \latex{10\times}€\latex{8=}€\latex{80}

Half price: \latex{10\times} (€\latex{8\div2) = 10\times} €\latex{4=}€\latex{40}

If one book is half price, then \latex{ 10 } books are also half price.

Changes in the product

Three children eat a total of \latex{ 12 } scoops of ice cream.

\latex{3\times 4=\textcolor{#ff0000}{12}}

Observe the changes in the product in the following cases.

When twice as many children eat the same number of scoops each:

\latex{(2\times 3)\times 4=}
\latex{=6\times 4=\textcolor{#ff0000}{12}}

If one factor is doubled (but the other is left unchanged), then the product is also doubled.

When the same number of children eat half the number of scoops each:

\latex{3\times (4\div2)=}
\latex{=3\times 2=\textcolor{#ff0000}{6}}

If one factor is halved (but the other is left unchanged),
then the product is also halved.

When twice as many children eat half the number of scoops each:

\latex{(2\times 3)\times (4\div2)=}
\latex{=6\times 2=\textcolor{#ff0000}{12}}

If one factor is doubled and the other is halved,
the product remains
the same.

Example

Do the following multiplications in the simplest way possible.
a) \latex{36\times 25;} b) \latex{68\times50;} c) \latex{33\times30.}

Solution

Based on what you've learnt about the changes of the product:

a) \latex{ 36 } \latex{ \times } \latex{ 25 } =

b) \latex{ 68 }\latex{ \times } \latex{ 50 } =

c) \latex{ 33 } \latex{ \times } \latex{ 30 } =

= \latex{ 9 } \latex{ \times } \latex{ 100 } = \latex{ 900 }

= \latex{ 34 } \latex{ \times } \latex{ 100 } = \latex{ 3,400 }

= \latex{ 99 } \latex{ \times } \latex{ 10 } = \latex{ 990 }

\latex{ \div4 }

x\latex{ 4 }

x\latex{ 2 }

\latex{ \div2 }

÷\latex{ 3 }

×\latex{3 }

Exercises

{{exercise_number}}. Do the multiplications as simply as possible.

a) \latex{80\times25}

b) \latex{50\times92}

c) \latex{125\times72}

d) \latex{400\times16}

{{exercise_number}}. Perform the following calculations. If you need help, take a look at the example task.

a) \latex{720\times 30}

b) \latex{47\times 20}

c) \latex{130\times 200}

d) \latex{250\times 40}

e) \latex{1,800\times 5}

f) \latex{76\times 50}

{{exercise_number}}. Which of the following statements are true and which are false

about a two-factor product?

a) The product is multiplied by three if one factor is multiplied by three but the other remains unchanged.

b) The product is unchanged if one factor increases while the other decreases.

c) The product is doubled if one factor is doubled while the other is unchanged.

d) The product is multiplied by four if both factors are doubled.

e) The product is multiplied by six if both factors are multiplied by three.

about a multiplication with multiple factors?

f) The product is unchanged if one of the factors is doubled, another is halved, and the rest are unchanged.

{{exercise_number}}. Do the multiplications as simply as possible.

a) \latex{2\times28\times5}

b) \latex{5\times57\times5\times4}

c) \latex{40\times9\times25}

d) \latex{50\times5\times7\times4\times5}

e) \latex{72\times18\times0\times25\times50}

{{exercise_number}}. Sylvia, Christie, and Melissa (from left to right in the image) have to multiply the numbers on their cardboards. Who is holding the largest product?

(All three were clever when counting.)

\latex{ 12 }
\latex{ 125 }
\latex{ 25 }
\latex{ 48 }

\latex{ 20 }
\latex{ 75 }
\latex{ 50 }
\latex{ 24 }

\latex{ 12 }
\latex{ 4 }
\latex{ 15 }
\latex{ 5 }
\latex{ 25 }
\latex{ 2 }
\latex{ 2 }

Quiz

Which is more? Half a dozen dozen dozen eggs or six dozen dozen eggs? (\latex{ 1 } dozen= \latex{ 12 } pieces.)

Mathematics 5.

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