GCSE Maths - What Does Inversely Proportional Mean? #91

00:04:34
https://www.youtube.com/watch?v=OCZSFmHnNI4

الملخص

TLDRThe video explores the concept of inverse proportionality through the scenario of farmers picking apples. This mathematical relationship shows that as the number of farmers (one variable) increases, the time taken to pick the apples (another variable) decreases proportionally. The relationship results in a downward sloping curve on graphs and can be formally defined using algebraic equations. Such equations require a constant of proportionality, noted as 'k'. For example, if the time taken is inversely proportional to the number of farmers, this can be expressed as t ∝ 1/f, with t as time and f as farmers. Adjustments in the number of farmers directly affect the time taken due to this constant. This useful tool in mathematics highlights proportional changes where increasing one element leads to a decreased rate in the other.

الوجبات الجاهزة

  • 🍏 Inversely proportional means one variable increases while the other decreases proportionally.
  • 📉 Graphs show inverse proportionality as a downward slope.
  • 🔢 Equation form includes a constant of proportionality.
  • 👩‍🌾 Example: More farmers reduce the apple-picking time.
  • ⬇️ Doubling one variable halves the other in inverse relations.
  • 🧮 Symbol '∝' denotes proportionality in equations.
  • 🔀 Structure: Variable equals constant divided by another variable.
  • 📝 Equations with additional numbers still count if structured properly.

الجدول الزمني

  • 00:00:00 - 00:04:34

    In the scenario of farmers picking apples, an inversely proportional relationship is discussed, where increasing the number of farmers decreases the time taken to pick the apples. This relationship is explained with the concept that as one variable increases, the other decreases proportionally, like when farmers double, the time halves. Such relationships can be represented graphically with a downward sloping curve or algebraically as equations, where time taken (t) is inversely proportional to the number of farmers (f), noted as t ∝ 1/f. By introducing a constant of proportionality (k), the relationship becomes t = k/f. The video further exemplifies the calculations using a specific value of k, demonstrating how to convert between time taken and number of farmers. Lastly, the video reiterates the concept of inverse proportionality, emphasizing that graphs of these relationships slope downwards and equations often take the form y = k/x.

الخريطة الذهنية

Mind Map

الأسئلة الشائعة

  • What is inversely proportional?

    Inversely proportional means as one variable increases, the other decreases proportionally.

  • How can inverse proportionality be represented graphically?

    It is represented as a downward sloping curve on a graph.

  • In the apple-picking example, what happens when the number of farmers doubles?

    The time taken to pick the apples will halve.

  • What does the equation for inverse proportionality include?

    It includes a constant of proportionality, usually represented by the letter k.

  • What is the algebraic expression for time taken inversely proportional to the number of farmers?

    It is expressed as t is proportional to 1/f, where t is time taken and f is the number of farmers.

  • How is the constant of proportionality denoted?

    It is usually denoted by the letter k in equations.

  • Does the specific value of the constant of proportionality matter?

    Yes, it can be any specific value depending on the relationship, and it affects calculations.

  • Can inverse proportionality include other numbers next to variables in an equation?

    Yes, as long as one variable equals something over another, it counts as inversely proportional.

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الترجمات
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التمرير التلقائي:
  • 00:00:00
    [Music]
  • 00:00:05
    let's imagine the scenario of farmers
  • 00:00:08
    picking apples from their orchard
  • 00:00:10
    and we're measuring how long it takes
  • 00:00:12
    them to pick all of the apples
  • 00:00:15
    now it's fairly obvious that the more
  • 00:00:17
    farmers there are hoping to pick the
  • 00:00:19
    apples the less time it will take to
  • 00:00:21
    pick them
  • 00:00:23
    in maths we call relationships like this
  • 00:00:25
    inversely proportional
  • 00:00:28
    which just means that as one of the
  • 00:00:30
    variables increases
  • 00:00:32
    the other variable decreases
  • 00:00:34
    proportionally
  • 00:00:36
    so in this case as the number of farmers
  • 00:00:38
    increases
  • 00:00:39
    the time taken decreases
  • 00:00:43
    and then the fact that is proportional
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    means that they increase and decrease at
  • 00:00:47
    the same rate
  • 00:00:49
    for example if the number of farmers
  • 00:00:51
    doubles
  • 00:00:52
    then the time taken will halve
  • 00:00:55
    or if we had 20 times as many farmers
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    then it would take 20 times less time to
  • 00:01:00
    pick the apples
  • 00:01:05
    next we need to look at how we can show
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    these sorts of relationships on graphs
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    and these will always have this sort of
  • 00:01:12
    downward sloping curve
  • 00:01:15
    if we start in the top left we can see
  • 00:01:17
    that this point corresponds to only a
  • 00:01:19
    few farmers
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    and a really long time taken
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    but as we move down into the right
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    like to this point down here
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    the number of farmers gets bigger and
  • 00:01:31
    bigger because we're moving to the right
  • 00:01:33
    and the time taken gets smaller and
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    smaller because we're moving down
  • 00:01:40
    another thing that we can do is express
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    the relationship as an algebraic
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    equation
  • 00:01:46
    the key thing to understand for this
  • 00:01:48
    is that saying that the time taken is
  • 00:01:51
    inversely proportional to the number of
  • 00:01:53
    farmers
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    is exactly the same thing as saying that
  • 00:01:56
    time taken is proportional to one over
  • 00:02:00
    the number of farmers
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    so if we let time taken be t
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    and the number of farmers be f
  • 00:02:07
    then we'd have t is proportional to one
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    over f
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    to write this as a proper equation
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    though we need to change the
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    proportional sign to an equal sign
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    and to do that we have to include a
  • 00:02:20
    constant of proportionality which we
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    normally show with the letter k
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    it's up to you where you put the k but
  • 00:02:27
    we normally put it in the top right of
  • 00:02:29
    the equation like we have here
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    now the particular value of k depends on
  • 00:02:34
    the specific relationship
  • 00:02:36
    so it could be anything
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    like 0.4
  • 00:02:40
    8 or whatever
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    if we pretend that in this case it was h
  • 00:02:44
    though
  • 00:02:45
    then as you can see we now have an
  • 00:02:47
    equation that we can use to convert
  • 00:02:49
    between time taken and the number of
  • 00:02:51
    farmers
  • 00:02:54
    for example if there were two farmers
  • 00:02:56
    then f would be two
  • 00:02:58
    so we could put the two into our
  • 00:03:00
    equation
  • 00:03:01
    and find that time would be equal to
  • 00:03:03
    eight divided by two
  • 00:03:05
    so it would take them four hours to pick
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    all of the apples
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    whereas if there were five farmers
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    then we'd do t equals eight over five 5
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    to find that it would only take 1.6
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    hours
  • 00:03:21
    so because we had more farmers in this
  • 00:03:22
    second case it took less time to pick
  • 00:03:25
    the apples
  • 00:03:29
    so to sum up this video
  • 00:03:31
    inversely proportional means that as one
  • 00:03:34
    variable increases
  • 00:03:35
    the other variable decreases
  • 00:03:37
    proportionally
  • 00:03:39
    and we can show these relationships on
  • 00:03:41
    graphs which will always have this
  • 00:03:43
    downward slope
  • 00:03:44
    or in equations
  • 00:03:46
    for example like y equals 2 over x
  • 00:03:49
    which will always have one variable
  • 00:03:51
    equals some number which we call our
  • 00:03:54
    constant of proportionality over the
  • 00:03:56
    other variable
  • 00:03:58
    and one last thing to point out is that
  • 00:04:00
    it doesn't matter if there are other
  • 00:04:01
    numbers next to our variables
  • 00:04:04
    like there's three or this five
  • 00:04:07
    as long as one variable is equal to
  • 00:04:09
    something over the other variable
  • 00:04:12
    then it counts as inversely proportional
  • 00:04:19
    anyway that's everything for this video
  • 00:04:21
    so hope you found it useful and cheers
  • 00:04:23
    for watching
  • 00:04:33
    you
الوسوم
  • inverse proportionality
  • mathematics
  • graphs
  • constants
  • equations
  • farmers
  • apples
  • proportional relationships