# Ratio Maths Problems with Solutions and Explanations for Grade 9

Detailed solutions and full explanations to ratio maths problems for grade 9 are presented.

 There are 600 pupils in a school. The ratio of boys to girls in this school is 3:5. How many girls and how many boys are in this school? Solution In order to obtain a ratio of boys to girls equal to 3:5, the number of boys has to be written as 3 x and the number of girls as 5 x where x is a common factor to the number of girls and the number of boys. The total number of boys and girls is 600. Hence 3x + 5x = 600 Solve for x 8x = 600 x = 75 Number of boys 3x = 3 � 75 = 225 Number of girls 5x = 5 � 75 = 375 There are r red marbles, b blue marbles and w white marbles in a bag. Write the ratio of the number of blue marbles to the total number of marbles in terms of r, b and w. Solution The total number of marbles is r + b + w The total ratio of blue marbles to the total number of marbles is r / (r + b + w) The perimeter of a rectangle is equal to 280 meters. The ratio of its length to its width is 5:2. Find the area of the rectangle. Solution If the ratio of the length to the width is 5:2, then the measure L of the length and and the measure W of the with can be written as L = 5x and W = 2x We now use the perimeter to write 280 = 2(2L + 2W) = 2(5x + 2x) = 14x Solve for x 280 = 14x x = 280 / 14 = 20 The area A of the rectangle is given by A = L � W = 5x � 2x = 10x2 = 10�202 = 4000 square meters The angles of a triangle are in the ratio 1:3:8. Find the measures of the three angles of this triangle. Solution If the ratio of the three angles is 1:3:8, then the measures of these angles can be written as x, 3x and 8x. Also the sum of the three interior angles of a triangle is equal to 180�. Hence x + 3x + 8x = 180 Solve for x 12x = 180 x = 15 The measures of the three angles are x = 15� 3x = 3 � 15 = 45� 8x = 8 � 15 = 120� The measures of the two acute angles of a right triangle are in the ratio 2:7. What are the measures of the two angles? Solution If the ratio of the two angles is 2:7, then the measures of two angles can be written as 2x and 7x. Also the two acute angles of a triangle is equal to 90�. Hence 2x + 7x = 90 9x = 90 x = 10 Measures of the two acute angles are 2x = 2 � 10 = 20� 7x = 7 � 10 = 70� A jar is filled with pennies and nickels in the ratio of 5 to 3 (pennies to nickels). There are 30 nickles in the jar, how many coins are there? Solution A ratio of pennies to nickels of 5 to 3 means that we can write the number of pennies and nickels in the form number of pennies = 5x and number of nickels = 3x But we know the number of nickels, 30. Hence 3x = 30 Solve for x x = 10 The total number of coins is given by 5x + 3x = 8x = 8 � 10 = 80 A rectangular field has an area of 300 square meters and a perimeter of 80 meters. What is the ratio of the length to the width of this field? Solution Let L and W being the length and the width (with L > W) of the rectangular field. The area and the perimeter are given; hence L � W = 300 (I) 2L + 2W = 80 (II) which is equivalent to L + W = 40 (III) We need to find the ratio L / W. Equation (I) gives W = 300 / L Substitute W by 300 / L in equation (III) L + 300 / L = 40 Multiply all terms in the above equation by L and simplify L2 + 300 = 40L Rewrite the equation in standard form, factor and solve L2 - 40 L + 300 = 0 (L - 10)(L - 30) = 0 Solutions: L = 10 and L = 30 We now calculate W For L = 10 , W = 300 / L = 300 / 10 = 30 m For L = 30 , W = 300 / L = 300 / 30 = 10 Since L > W, we select the soultion L = 30 and W = 10 and the L / W is equal to 30 / 10 = 3 / 1 or 3:1 Express the ratio 3 2/3 : 7 1/3 in its simplest form. Solution We first convert the mixed numbers 3 2/3 and 7 1/3 into fractions 3 2/3 = 3+ 2 / 3 = 3 � 3 / 3 + 2 / 3 = 9 / 3 + 2 / 3 = 11 / 3 7 1/3 = 7 + 1 / 3 = 7 � 3 / 3 + 1 / 3 = 22 / 3 The ratio 3 2/3 : 7 1/3 can be expressed as 11 / 3 � 22 / 3 = 11 / 3 � 3 / 22 Simplify = 11 / 22 = 1 / 2 The ratio is 1 / 2 or 1:2 The length of the side of square A is twice the length of the side of square B. What is the ratio of the area of square A to the area of square B? Solution Let x be the length of the side of square A and y be the length of the side of square B with x = 2 y. Area of A and B are given by A = x2 and B = y2 But x = 2y. Hence A = (2y)2 = 4 y2 The ratio of A to B is 4 y2 / y2 = 4 / 1 or 4:1 The length of the side of square A is half the length of the side of square B. What is the ratio of the perimeter of square A to the perimeter of square B? Solution Let 2 x be the side of square B and x be the side of square A (half). Perimeter of square A = 4 x , perimeter of square B = 4 (2 x) = 8 x The ratio R of the perimeter of A to the perimeter of B is R = 4 x / 8 x = 1 / 2 At the start of the week a bookshop had science and art books in the ratio 2:5. By the end of the week, 20% of each type of books were sold and 2240 books of both types were unsold. How many books of each type were there at the start of the week? Solution Let S and A be the number of science and art books respectively at the start of the week. Hence S / A = 2 / 5 If 20% of each type of books were sold then 80% of each were unsold at the end of the week and their total is known: 2240. Hence 80% S + 80 % A = 2240 or 0.8 S + 0.8 A = 2240 We now need to solve the two equations in S and A obtained above to find the number of books of each type. Use the cross product on the equation S / A = 2 / 5 to obtain 5 S = 2 A Solve the system of equations 0.8 S + 0.8 A = 2240 and 5 S = 2 A to obtain S = 800 and A = 2000 At the start of the month a shop had 20-inches and 40-inches television sets in the ratio 4:5. By the end of the month, 200 20-inches and 500 40-inches were sold and the ratio of 20-inches to 40-inches television sets became 1:1. How many television sets of each type were there at the start of the month? Solution Let x and y be the number of 20-inches and 40-inches television sets respectively at the start of the month. Hence x / y = 4 / 5 By the end of the month 200 and 500 were sold from the 20-inches and 40-inches television sets respectively. Therefore x - 200 and y - 500 were unsold and their ratio is known and equal to 1:1. Hence (x - 200) / (y - 500) = 1 / 1 Use the cross product on both equations obtained above 5x = 4 y and x - 200 = y - 500 Solve the system of equations to obtain x = 1200 and y = 1500 The aspect ratio of a tv screen is the ratio of the measure of the horizontal length to the measure of the vertical length. Find the horizontal length and vertical height of a tv screen with an aspect ratio of 4:3 and a diagonal of 50 inches. Solution Let H be the horizontal length and V be the vertical height of the tv. Their ratio is given. Hence H / V = 4 / 3 or use cross product to obtain: 3 H = 4 V The relationship between the horizontal length, the vertical height and the diagonal is given by Pythagora's theorem as follows: H 2 + V 2 = 50 2 We now solve the system of equations 3 H = 4 V and H 2 + V 2 = 50 2 to obtain H = 40 and V = 30