# 6th Grade - Cross-Multiply To Solve Proportions

 Grade Level: 6th Skill: Fractions, Decimals and Percentages Topic: Cross-Multiply to Solve Proportions Goal: Use proportions to solve problems (e.g., determine the value of N if 4 - 7 = N - 21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplicatio Skill Description: Recognize that two equivalent ratios can be written as a proportion. (a/b=c/d). Use proportions to solve problems by either finding a common number that can be multiplied or divided by both terms of a ratio to find a missing value in the second ratio or using cross multiplication.

### Sample Problems

 (1) 2/5 = 4/n (n = 10) (2) 100 pages in 30 minutes = ________ pages in 15 minutes (50 pages) (3) The height of a one story building is 15 feet, with a shadow the length of 30 feet. The height of a tree is 12 feet. What would the length of the shadow of the tree be? (n = 6) (4) Triangle A and Triangle B are similar. Triangle A has the following measurements: side one = 2 cm, side two = 4 cm, side three =2 cm. If Triangle B has side one – 4 cm, what would be the measure of side two? (side 2 = 8 cm) (5) Stacy can type 50 words per minute. How many words could she type in one hour? (Remember, there are 60 minutes in an hour.) (3,000 words)

### Learning Tips

 (1) Remind students that proportion is simply the name used to describe two ratios that are equal. This is true when both terms of the ratio are either multiplied or divided by the same number. (2) To determine if two ratios are proportional (equivalent) you must follow three steps. First, compare the units to make sure they’re in the same place on each ratio (i.e. 4 laps/8minutes = 16 laps/32minutes). Both ratios have laps as the first term and minutes as they second term, so it could be a proportion. Next, write each ratio in simplest form. To do this you need to divide each term of the ratio by the GCF, greatest common factor. The GCF is the largest number that will divide into both terms evenly without remainders. For example, the GCF of 4/8 is 4 because 4 is the largest number that both terms are divisible by. Hence, 4/8 in simplest form is ½ because 44=1 and 84=2. Lastly, compare the simplest forms of both ratios. If they’re the same that the ratios are proportional. The simplest form of 4/8 is ½. The simplest form of 16/32 is ½. Since both ratios are ½ in simplest form they are proportional. (3) Cross multiplication is a short cut to solve proportions by multiplying the diagonals. It is called cross multiplication because you multiply ratio terms diagonally, here by making an x or cross shape. Always use the fraction form of a ratio to use cross multiplication to solve proportions. To cross multiply you multiply the top term of the first ratio by the bottom term of the second ratio. Write the product you get in front of an equal sign. For example, if you have 5/25 = n/50 you would first multiply 5x50 to get 100. Next, multiply the bottom term of the first ratio by the top term of the second ratio. So, 25xn. Your new problem would be 100=25n. To complete the problem use steps to solve algebraic equations: 10025=4, so n=4. (4) Have children use two different colors of ink to draw the diagonals as they cross multiply. For example, a student can use a red marker to connect the top term from ratio 1 to the bottom term of ratio two and write that product in red. He/she can then use a blue pen to draw a line across the diagonal from the bottom term from ratio 1 to the top term of ratio 2 and write the product they make next to the red product. Next, a student can draw and equal sign between them in regular pencil and begin solving. Using different colored pens keeps students focused on each step one at a time and prevents errors in choosing the incorrect terms to multiply. (5) Inform students that it is helpful to solve proportions to find the best deal. For example, if they can buy one cd for \$7.50 or 5 for \$35.00, they could use a proportion to determine if it is a better deal to buy the 5. (6) Remind students that the last step to solve a proportion will always be to divide. (7) When solving word problems with proportions, have students create a proportion using “word ratios” before attempting to solve. Sample Problem: The length of the shadow from a tree in Johnny’s yard is 12 feet. Johnny is 5 feet tall and his shadow is 10 feet. What is the height of the tree? height of tree height of Johnny __________________ = _________________ length of shadow length of shadow As in the word ratio above, it is crucial that the same unit of measure be on numerator (top) of each ratio. Hence, height is the word on the top of ratio one and the top or ratio two. The same goes for the denominator. They must always be the same unit. The most commonly made mistake of students is incorrect placement of terms.

### Online Resources

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### Extra Help Problems

 (1) 2/5 = n/35 (n = 14) (2) 11/33 = 55/g (g = 165) (3) 3/6 = 1/a (a = 2) (4) 7/35 = b/5 (b = 1) (5) 9/5 = n/11 (n = 19.8) (6) 2/3 = 72/a (a = 108) (7) a/5 = 16/8 (a = 10) (8) 14/b = 1/16 (b = 224) (9) 26/13 = 8/n (n = 4) (10) s/12 = 30/15 (s = 24) (11) Tell whether the ratios are proportional and explain why or why not. 9/11 = 18/22 (The ratios are proportional, GCF = 2.) (12) Tell whether the ratios are proportional and explain why or why not. 11:5 = 44:15 (The ratios are not proportional, there is no GCF.) (13) Tell whether the ratios are proportional and explain why or why not. 120/2 = 1/60 (Not proportional, no GCF) (14) Tell whether the ratios are proportional and explain why or why not. 3/9 = 2/18 (Not proportional, no GCF) (15) Tell whether the ratios are proportional and explain why or why not. 140 to 7 = 20 to 1 (Proportional, GCF = 7) (16) 6 feet in 3 minutes = 18 feet in ______ minutes (9 minutes) (17) 60 miles per hour = ________ miles in 4 hours (240 miles) (18) 16 oz. in one pound = ________ ounces in 98 pounds. (1,568 ounces) (19) _________ words per minute = 150 words per 15 minutes (10 words per minute) (20) Kenneth is building a model of his dream car. The length of the front of his model is 17 cm. The height is 35 centimeters. If Kenneth wanted to build a model that was 10 times larger, what would the length and height measure? (Length = 170 cm, height = 350 cm) (21) The scale on a map is 1 inch = 25 miles. If Minh measures the distance from his house to his best friend’s house and it is 5 inches on the map, what is the distance between Minh’s house and his best friend’s house? (125 miles) (22) Tanya can email 10 friends in 30 minutes. She has 50 friends she’d like to email. How long will it take her? (150 min or 2 ½ hrs.) (23) Jasmin can run 4 miles in 24 minutes. How fast does she run a mile? (6 min per mile) (24) Jamie bought 8 CDs for \$5.84. Shane bought 30 CDs for \$18.50. Who got the best deal? (Shane, he paid \$0.62 per CD. Jamie paid \$0.73 each.) (25) Kylie went to the store to buy cupcakes for her class. She sees that she can buy sprinkled cupcakes for \$1.25 each or frosted cupcakes for 5 for \$7.50. Which will save her the most money? (The sprinkled cupcakes are a better deal, because the frosted cupcakes work out to be \$1.50 each.)