Question

1. the perimeter of a rectangle is 280 cm. the ratio of the width to the length is 3 : 4. what is the length of the rectangle?
2. the ratio of country albums to jazz albums in a music collection is 2 : 3. if the music collection has 45 albums, how many are country albums?
3. the lengths of the sides of a triangle are in the extended ratio 3 : 6 : 8. the triangle’s perimeter is 510 cm. what are the lengths of the sides?

Explanation:

Problem 6

Step1: Recall perimeter formula of rectangle

The perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \), where \( l \) is length and \( w \) is width. Given \( P = 280 \) cm, so \( 2(l + w)=280 \), which simplifies to \( l + w = 140 \) cm.

Step2: Use the ratio to define variables

The ratio of width to length is \( 3:4 \). Let width \( w = 3x \) and length \( l = 4x \), where \( x \) is a positive real number.

Step3: Substitute into the sum of length and width

Substitute \( w = 3x \) and \( l = 4x \) into \( l + w = 140 \). We get \( 3x + 4x=140 \), which is \( 7x = 140 \).

Step4: Solve for x

Divide both sides by 7: \( x=\frac{140}{7} = 20 \).

Step5: Find the length

Since length \( l = 4x \), substitute \( x = 20 \), we get \( l=4\times20 = 80 \) cm.

Step1: Determine the total parts of the ratio

The ratio of country albums to jazz albums is \( 2:3 \). So total parts \( = 2 + 3=5 \) parts.

Step2: Find the value of one part

The total number of albums is 45. So value of one part \(=\frac{45}{5}=9 \) albums per part.

Step3: Calculate the number of country albums

Country albums are 2 parts. So number of country albums \( = 2\times9 = 18 \).

Step1: Define the sides using the ratio

The sides of the triangle are in the ratio \( 3:6:8 \). Let the sides be \( 3x \), \( 6x \), and \( 8x \), where \( x \) is a positive real number.

Step2: Recall the perimeter formula of a triangle

The perimeter \( P \) of a triangle is the sum of its three sides. Given \( P = 510 \) cm, so \( 3x+6x + 8x=510 \).

Step3: Combine like terms

Combine the left - hand side: \( 17x = 510 \).

Step4: Solve for x

Divide both sides by 17: \( x=\frac{510}{17}=30 \).

Step5: Find the lengths of the sides

• First side: \( 3x=3\times30 = 90 \) cm.
• Second side: \( 6x = 6\times30=180 \) cm.
• Third side: \( 8x=8\times30 = 240 \) cm.

Answer:

The length of the rectangle is \( 80 \) cm.

Problem 7