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AMC 10 USA Math Olympiad

Average Problem – AMC 10B – 2019 – Problem No – 4

Average – AMC 10B – 2019 – Problem No – 4


Let’s try this problem based on average from AMC 10B, 2019.

Mr. Patrick teaches math to 15  students. He was grading tests and found that when he graded everyone’s test except Payton’s, the average grade for the class was 80 . After he graded Payton’s test, the test average became 81. What was Payton’s score on the test?

  • 81
  • 95
  • 85
  • 91

Key Concepts


Average

Mean

Arithmetic

Check the Answer


Answer: 95

AMC 10A – 2015 – Problem No – 5

Challenges and Thrills in Pre-College Mathematics

Try with Hints


If you are not getting correct answer you can start from here :

The average of a set of numbers is the value we get if we evenly distribute the total across all entries. So assume that the first  14 students each scored  80.

If Payton also scored an 80 the average would still be 80. In order to increase the overall average to 81 we need to add one more point to all of the scores, including Payton’s. This means we need to add a total of  15 more points, so Payton needs 80+15 = 95 .

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AMC 10 USA Math Olympiad

Average Problem from AMC 10A – 2020 -Problem No. 6

What is Average ?


In mathematics and statistics, average refers to the sum of a group of values divided by n, where n is the number of values in the group. An average is also known as a mean.

Try this sum from AMC 10 – 2020


Driving along a highway, Megan noticed that her odometer showed 15951 (miles). This number is a palindrome-it reads the same forward and backward. Then 2 hours later , the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this 2 – hour period ?

A) 50 B) 55 C)60 D) 65 E) 70

American Mathematics Competition 10 (AMC 10B), {2020}, {Problem Number 6}

Average

4 out of 10

Mathematics can be fun

Knowledge Graph


Average- knowledge graph

Use some hints


Do you really need any hint ???

Try this out:

In order to get the smallest palindrome greater than 15951 , we need to raise the middle digit. If we were to raise any of the digits after the middle, we would be forced to also raise a digit before the middle to keep it a palindrome, making it unnecessarily larger.

So what can we do here ?

We can raise 9 to the next largest value, 10 , but obviously, that’s not how place value works, so we’re in the 16000 s now . To keep this a palindrome, our number is now 16061.

If you really need the final hint this can be the life saver for this sum :

So Megan drove 16061 – 15951 = 110 miles . Since this happened over 2 hours , she drove at \(\frac {110}{2}\) = 55 mph .

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AMC 8 USA Math Olympiad

Mean and Median calculation AMC 8, 2013 Problem 5

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What are we learning ?

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Competency in Focus: Mean and Median calculation

This problem from American Mathematics Contest 8 (AMC 8, 2013) is based on calculation of mean and median. It is Question no. 5 of the AMC 8 2013 Problem series.

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First look at the knowledge graph:-

[/et_pb_text][et_pb_image src=”https://www.cheenta.com/wp-content/uploads/2020/01/AMC-8-2013-Problem-5-1.png” alt=”calculation of mean and median- AMC 8 2013 Problem” title_text=” mean and median- AMC 8 2013 Problem” align=”center” force_fullwidth=”on” _builder_version=”4.2.2″ min_height=”429px” height=”189px” max_height=”198px” hover_enabled=”0″ custom_padding=”10px|10px|10px|10px|false|false”][/et_pb_image][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″ inline_fonts=”Aclonica”]

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″]Hammie is in the $6^\text{th}$ grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?  [/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” _builder_version=”4.1″ open=”on”]American Mathematical Contest 2013, AMC 8 Problem 5

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Basic Statistics and Data Representation mainly calculation of mean and median.

[/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.1″ open=”off”]4/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” open=”off” _builder_version=”4.0.9″]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″][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″]

Let us first find the median of the weight of the five children. For this, we first have to arrange the weights of the five children in increasing order. As we know, the median is the middle value, if there is an odd number of observations, and if there is an even number of observations, it is the average of the two middle values. Thus, lining up the numbers (5, 5, 6, 8, 106), we see that it  is 6 pounds.

[/et_pb_tab][et_pb_tab title=”HINT 2″ _builder_version=”4.1″]Now what we have to find is the mean of the weights of five children .The average weight of the five kids is $\dfrac{5+5+6+8+106}{5} = \dfrac{130}{5} = 26$.[/et_pb_tab][et_pb_tab title=”HINT 3″ _builder_version=”4.1″]The median here is obviously less than the mean.[/et_pb_tab][et_pb_tab title=”HINT 4″ _builder_version=”4.1″]Therefore, the average weight is bigger than median weight , by $26-6 = 20$ pounds, making the answer , average by 20.[/et_pb_tab][/et_pb_tabs][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built=”1″ fullwidth=”on” _builder_version=”4.2.2″ global_module=”50833″][et_pb_fullwidth_header title=”AMC – AIME Program” button_one_text=”Learn More” button_one_url=”https://www.cheenta.com/amc-aime-usamo-math-olympiad-program/” header_image_url=”https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png” _builder_version=”4.2.2″ background_color=”#00457a” custom_button_one=”on” button_one_text_color=”#44580e” button_one_bg_color=”#ffffff” button_one_border_color=”#ffffff” button_one_border_radius=”5px”]

AMC – AIME – USAMO Boot Camp for brilliant students. Use our exclusive one-on-one plus group class system to prepare for Math Olympiad

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