Mobile Math Teachers’ Circle
Ratio and Proportion
June 15, 2016
Session for 6th Grade Teachers
1.
(a) Every bird has two wings and one beak. What is the ratio of wings to beaks in
the bird house at the zoo?
(b) At the last election candidate A received 210 votes, candidate C received only 70 votes.
For every vote for candidate C, how many votes did candidate A receive?
(c) We paid $75 for 15 hamburgers. How much did we pay per hamburger? How much would
20 hamburgers cost?
2.
It took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed
in 35 hours?
¯ CE,
¯ EG,
¯ and
3.
In square ACEG shown below, B, D, F , and H are midpoints of AC,
¯ respectively. A student thinks that the area of the triangle ∆DEF , shown shaded, is 1
AG,
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the area of square ACEG. Explain why the student is NOT correct.
C
D
B
A
E
F
H
G
4.
Kate is placing her new bookcase against a wall in her bedroom. The height of the
bookcase is 84 inches and the width is one third of the height. What is the area, in square
inches, of the section of the wall the bookcase will cover?
5.
Matt has a piano book with 84 songs in it. Matt has learned 25% of these songs.
What is the total number of songs in this book that Matt has NOT learned to play?
6.
How many of the light coins do we have to remove, such that exactly 90% of the remaining coins are light?
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iiii
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7. Today people spend a lot of time sitting in front of televisions, computers, smart phones,
or other electronic devices with screens. Have you ever noticed that not all screens are the
same rectangle? That is to say, they are not proportional or similar. The height to width ratio
is not the same when you start comparing analog tube televisions, HD televisions and movie
theatre screens. If you have been to an electronics store recently you have noticed many similar
figures showcased along a wall. I am talking about HD televisions. HD screens are all similar
to each other. The height to width ratio of HD screens is always the same. The same can be
said for analog tube televisions and for most movie theatre screens and movie formats.
The typical HD screen has a width to height ratio of 16 to 9.
Find the missing dimension for each of my relatives’ HD screens. Assume that all dimensions
are in inches.
(a) The screen of my parents’ HD television has a width of 24 inches.
(b) Aunt Edna’s screen is 12 inches high.
(c) Uncle Bob has a big one. His screen is 48 inches wide.
Movie theatre screens come in the widescreen format. The ratio of width to height is 2.35 to 1.
(d) What are the dimensions of the movie screen at the local theatre if its width is 37.6 feet?
(e) Some movies are shown in the widescreen format on television. In order to fit the screen
black bars are use at the top and the bottom of the television screen. I want to watch a movie
in widescreen format on Uncle Bob’s HD screen. What are the dimensions of the black bars
on the top and the bottom?
8. The other day I stopped for gas and I saw the sign below, attached to the soda cooler.
(a) I wondered which size soda was the best deal? Make a guess and explain your thoughts.
(b) Use the information from the picture to fill in the first two columns of the table then find
the price per ounce for each bottle. Consider any rounding of ounces or price that you might
do to make “nicer” numbers.
Ounces of Soda (oz.)
Price ($)
Price per Ounce ($)
(c) Which size of soda do you think is the best deal? How do you know?
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9. In the illustration below, the two scales are balanced. How many “C” balls does it take to
balance one “A” ball?
A
A
A
A
A
B
B
B
B
A
C
C
10. (AJHSME 1997) At the grocery store last week, small boxes of facial tissue were priced
at 4 boxes for $5. This week they are on sale at 5 boxes for $4. What is the percent decrease
in the price per box during the sale?
11.
(AJHSME 1997) Three bags of jelly beans contain 26, 28, and 30 beans. The ratio
of yellow beans to all beans in each of these bags are 50%, 25%, and 20%, respectively. All
three bags of candy are dumped into one bowl. What is the ratio of yellow jelly beans to all
beans in the bowl?
12.
(AJHSME 1998) At Annville Junior High School, 30% of the students in the Math
Club are in the Science Club, and 80% of the students in the Science Club are in the Math
Club. There are 15 students in the Science Club. How many students are in the Math Club?
Sources:
Problems 1 through 3, Grade 6 Overview, 2015 Revised Alabama Course of Study: Mathematics.
Problems 4 and 5, Exemplar Grade 6 Mathematics Test Questions, ACT Aspire,
https://www.discoveractaspire.org/
Problem 6, Känguru der Mathematik, http://www.mathe-kaenguru.de/wettbewerb/index.
html Problem 7, http://www.yummymath.com/2015/glowing-rectangles-2/
Problem 8, http://www.yummymath.com/2014/an-ounce-of-cola/
Problem 9, The Great Math Experience: Engaging Problems for Middle School Mathematics, by Andrew Field, Publisher: Trafford Publishing (May 18, 2004), ISBN-10: 141201509X,
ISBN-13: 978-1412015097.
The American Mathematics Contest 8 (AMC 8) (formerly the American Junior High School
Mathematics Examination AJHSME) is a mathematics competition for students in grades 8
and below.
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