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June 18, 2019

Ratio with the sum of digits

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Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.22.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px"]Find the 3-digit number whose ratio with the sum of its digits is minimal.

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.22.4"][et_pb_column type="4_4" _builder_version="3.22.4"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.23.3" 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="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]

Albania TST 2013 [/et_pb_accordion_item][et_pb_accordion_item title="Topic" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Number Theory, Inequalities. [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Easy [/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Problem Solving Strategies by Arthur Engel [/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]

Start with hints

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.23.3"]Suppose that the number is (abc)_{10}. Then we would like to minimise \frac{100a+10b+c}{a+b+c}. Try to minimise for one variable at a time. [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.23.3"]Note that, \frac{100a+10b+c}{a+b+c}=1+\frac{9(11a+b)}{a+b+c}. In this expression, it is possible to minimise for c independent of a,b. [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.23.3"]From the previous hint, the expression is minimised for c=9. Show that the ratio can now be rewritten as 1+\frac{10a-9}{a+b+9}. Minimise for $b$. [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.23.3"]In the previous hint we see that the minimising value of b is also 9. Finally, the ratio may be written as 10-\frac{189}{a+18}. This is minimised for a=1. Hence the answer is 199. [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]

Watch the video (Coming Soon)

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Similar Problems

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2 comments on “Ratio with the sum of digits”

  1. ratio can now be rewritten as 1+\frac{10a-9}{a+b+9}. Minimise for $b$.; I do not understand it

    1. From the previous hint, the expression is minimised for c=9. Show that the ratio can now be rewritten as 1+\frac{10a-9}{a+b+9}. Minimise for $b$.

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