Median of numbers | AMC-10A, 2020 | Problem 11

Try this beautiful problem from Geometry based on Median of numbers from AMC 10A, 2020.

Median of numbers - AMC-10A, 2020- Problem 11


What is the median of the following list of $4040$ numbers$?$

\(1,2,3,.......2020,1^2,2^2,3^2...........{2020}^2\)

  • $1989.5$
  • $ 1976.5$
  • $1972.5$

Key Concepts


Median

Algebra

square numbers

Check the Answer


Answer: $1976.5$

AMC-10A (2020) Problem 11

Pre College Mathematics

Try with Hints


To find the median we need to know how many terms are there and the position of the numbers .here two types of numbers, first nonsquare i.e (1,2,3......2020) and squares numbers i.e \((1^2,2^2,3^2......2020^2)\).so We want to know the \(2020\)th term and the \(2021\)st term to get the median.

Can you now finish the problem ..........

Now less than 2020 the square number is \({44}^2\)=1936 and if we take \({45}^2\)=2025 which is greater than 2020.therefore we take the term that \(1,2,3...2020\) trms + 44 terms=\(2064\) terms.

can you finish the problem........

since $44^{2}$ is $44+45=89$ less than $45^{2}=2025$ and 84 less than 2020 we will only need to consider the perfect square terms going down from the 2064 th term, 2020, after going down $84$ terms. Since the $2020$th and $2021$st terms are only $44$ and $43$ terms away from the $2064$th term, we can simply subtract $44$ from $2020$ and $43$ from $2020$ to get the two terms, which are $1976$ and $1977$. Averaging the two,=\(1976.5\)

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Area of Triangle Problem | AMC-8, 2019 | Problem 21

Try this beautiful problem from Geometry based on the area of the triangle.

Area of Triangle - AMC-8, 2019- Problem 21


What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$

  • $8$
  • $ 16$
  • $15$

Key Concepts


Geometry

Triangle

Linear equation

Check the Answer


Answer: $16$

AMC-8 (2019) Problem 21

Pre College Mathematics

Try with Hints


Find the three vertex of the triangle

Can you now finish the problem ..........

The area of the Triangle =\(\frac{1}{2} \times \{x_1(y_2 - y_3)+x_2(y+3 -y_1)+x_3(y_1 -y_2)\}\)

can you finish the problem........

Area of the triangle

Solving two The lines y=5 and y=1+x are intersect at (4,5)=\((x_1,y_1)\)(say)

Solving two The lines y=5 and y=1-x are intersect at (-4,5)=\((x_2,y_2)\)(say)

Solving two The lines y=1-x and y=1+x are intersect at (0,1)=\((x_1,y_1)\)(say)

Then the area of Triangle =\(\frac{1}{2} \times\{ x_1(y_2 - y_3)+x_2(y+3 -y_1)+x_3(y_1 -y_2)\}\)

= \(\frac{1}{2} \times \{4(5-1)+(-4)(1-5)+0(5-5)=\frac{1}{2} (16+16)=16\}\)

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Linear Equation AMC 8 2010 problem 21

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What are we learning ?

[/et_pb_text][et_pb_text _builder_version="4.1" text_font="Raleway||||||||" text_font_size="20px" text_letter_spacing="1px" text_line_height="1.5em" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Competency in Focus: Linear Equation This problem from American Mathematics contest (AMC 8, 2010 problem 21) is based on Linear Equation[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

First look at the knowledge graph.

[/et_pb_text][et_pb_image src="https://www.cheenta.com/wp-content/uploads/2020/01/AMC-8-problem-21.png" align="center" force_fullwidth="on" _builder_version="4.1" custom_margin="||||false" custom_padding="20px||20px||false|false"][/et_pb_image][et_pb_video _builder_version="4.1"][/et_pb_video][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Next understand the problem

[/et_pb_text][et_pb_text _builder_version="4.1" text_font="Raleway||||||||" text_font_size="20px" text_letter_spacing="1px" text_line_height="1.5em" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read \( \frac{1}{5}\) of the pages plus 12 more, and on the second day she read 1/4 of the remaining pages plus 15 pages. On the third day she read 1/3 of the remaining pages plus 18 pages. She then realized that there were only 62 pages left to read, which she read the next day. How many pages are in this book? (A) 120  (B) 180  (C) 240  (D) 300  (E) 240[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="4.0"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.1" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.1"]American Mathematical Contest 2010, AMC 8 Problem 21 [/et_pb_accordion_item][et_pb_accordion_item title="Key Competency" _builder_version="4.1" open="off"]This number theory problem is based on the concept of linear equation[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.1" open="off"]5/10[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.1" open="off"]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

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Start with hints 

[/et_pb_text][et_pb_tabs _builder_version="4.1" hover_enabled="0"][et_pb_tab title="HINT 0" _builder_version="4.0.9"]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title="HINT 1" _builder_version="4.1" hover_enabled="0"]You can assume total number of pages to be a variable say x. [/et_pb_tab][et_pb_tab title="HINT 2" _builder_version="4.1"]Every next day she is reaing a fraction of the REMAINING pages, not the total pages.[/et_pb_tab][et_pb_tab title="HINT 3" _builder_version="4.1"]she also reads some more pages on each day, so we need to subtract this also to get the actual remaining page[/et_pb_tab][et_pb_tab title="HINT 4" _builder_version="4.1"]At the en of third day total remining page in term of x must be 62 as state in the question.[/et_pb_tab][et_pb_tab title="HINT 5" _builder_version="4.1"][/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

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