Short Trick to bring given set of numbers in proportion
Duration: 14 min
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AI Summary
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This educational video provides a comprehensive lesson on Ratio and Proportion, focusing on a specific problem type: finding a number to add or subtract from a set of four numbers to make them proportional. The instructor begins by defining a ratio as a comparison of two quantities and illustrating it with visual aids like a colorful pie chart and a slide titled 'Ratio a comparison of 2 quantities.' The core of the lecture involves solving the problem 'What should be added to each of 5, 9, 11, 18 to make them in proportion?' using two distinct methods. First, a standard algebraic approach is demonstrated where a variable 'x' is added to each term, leading to a quadratic equation that simplifies to a linear one. Second, a shortcut formula is introduced: |ad - bc| / |(a+d) - (b+c)|. This method is applied to a second, more complex set of numbers (19, 37, 47, 86) to demonstrate its efficiency. Finally, the lesson explores a variation where a number must be subtracted, showing how a negative result in the algebraic method correctly implies addition. The video uses a consistent orange background with space-themed graphics (rockets, planets) to maintain engagement throughout the mathematical derivations.
Chapters
0:00 – 2:00 00:00-02:00
The video opens with a title card 'RATIO & PROPORTION' featuring a 3D pie chart graphic. The instructor introduces the topic, defining a ratio as 'a comparison of 2 quantities.' A slide appears showing the text 'Ratio a comparison of 2 quantities' alongside a visual representation of parts and a whole using colored dots. The instructor mentions that ratios can be written in three ways: colon, word, or fraction form. This section establishes the foundational definitions before moving to problem-solving, using a pink background with space-themed doodles like rockets and planets. The instructor is visible in the bottom right corner throughout.
2:00 – 5:00 02:00-05:00
The instructor presents the first problem: 'What should be added to each of 5, 9, 11, 18 to make them in proportion?' He sets up the proportion as (5+x) : (9+x) :: (11+x) : (18+x). He converts this to the fractional equation (5+x)/(9+x) = (11+x)/(18+x). Through cross-multiplication, he derives (5+x)(18+x) = (11+x)(9+x). Expanding the terms yields 90 + 18x + 5x + x^2 = 99 + 9x + 11x + x^2. After canceling x^2 and rearranging, he solves 3x = 9 to find x = 3. He verifies the result by showing the new ratios 8:12 and 14:21 both simplify to 2:3, confirming the proportionality. He underlines key terms on the screen to emphasize the steps.
5:00 – 10:00 05:00-10:00
A shortcut method is introduced for this specific problem type. The instructor writes the numbers 5, 9, 11, 18 and calculates the cross products of the extremes (5*18=90) and means (9*11=99). The difference is 9. He then calculates the difference between the sum of extremes (5+18=23) and the sum of means (9+11=20), which is 3. Dividing the first difference by the second (9/3) gives the answer 3. He applies this formula to a new problem: 'What should be added to each of 19, 37, 47, 86?' He calculates 19*86=1634 and 37*47=1739, finding a difference of 105. The sum difference is (19+86) - (37+47) = 105 - 84 = 21. Dividing 105 by 21 yields 5. He writes the formula |ad-bc| / |(a+d)-(b+c)| on the board to formalize the shortcut.
10:00 – 13:47 10:00-13:47
The lesson shifts to a variation: 'What should be subtracted from each of 5, 9, 11, 18 to make them in proportion?' The instructor sets up the algebraic equation (5-y)/(9-y) = (11-y)/(18-y). Cross-multiplying gives (5-y)(18-y) = (11-y)(9-y). Expanding leads to 90 - 5y - 18y + y^2 = 99 - 11y - 9y + y^2. Simplifying results in -3y = 9, so y = -3. The instructor explains that subtracting a negative number is equivalent to adding a positive number, so the answer is effectively +3. He circles '+3' on the screen to conclude the problem. The video ends with a 'THANKS FOR WATCHING' slide.
The video effectively bridges theoretical definitions with practical problem-solving techniques. It starts with basic definitions of ratio and proportion, then moves to a standard algebraic solution for finding a common term to add to a sequence. Crucially, it introduces a faster, formula-based shortcut method (|ad-bc| / |(a+d)-(b+c)|) which is demonstrated on two different sets of numbers. The lesson concludes by handling a subtraction scenario, showing how algebraic results (negative values) translate back to the physical operation (addition), reinforcing the consistency of the mathematical logic. The consistent use of visual aids and step-by-step board work ensures clarity for students learning these concepts.