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This represents the fraction of the bottle that is full before class. This means that Paulina drinks ten-twelfths of two-thirds of the bottle of water during class. To find this amount, these fractions will be multiplied. Expression [0.5em] 10/12 * 2/3 A number line can be used to help with this concept. First, divide the number line between 0 and 1 into thirds. Fill in 23, representing the water that remains in the bottle before class. Then, divide this section into 12 smaller equal parts and color in 10 of them.

Notice that each third is divided into six smaller equal parts. The other third can also be divided into six parts. This will make it easier to find what fraction of the whole the red part represents.

The red part shows what fraction of the water bottle Paulina drinks during class.

10/12 * 2/3 Now compare each fraction with 12. These fractions do not have the same denominator. The denominators of the fractions are 12, 3, and 2. The least common denominator of these numbers is 12. Each fraction can be rewritten as an equivalent fraction with the common denominator to make it easier to compare them.

	Rewrite	Compare with 12, or 612
10/12	10/12	10/12 > 6/12
2/3	2*4/3*4 = 8/12	8/12 > 6/12

Both fractions are greater than 12. An estimate for the product can then be 1 because the fractions can be rounded to 1. rccc Product: & 10/12 & * & 2/3 [0.5em] & ↓ & & ↓ Estimate: & 1 & * & 1 Since 1 is the identity element of multiplication, this product is also equal to 1. However, that does not say much about the original product, and 1 is not one of the possible answer options. Now think of rounding only one of the fractions to 1 one at a time.

	Estimate for 1012* 23
Round 1012 to 1	1 * 2/3 = 2/3
Round 23 to 1	10/12 * 1 = 10/12

The fraction 1012 can be simplified to 56. The answer is either 23 or 56. The fractions in the options are in simplest form and 23 is among the options. Therefore, the answer is 23.

Notice that the numerator and denominator both have 4 as a factor. This suggests that the resulting fraction can be reduced to its lowest form by dividing the numerator and denominator by this factor. The product of the fractions is 59. This is the fraction of the bottle of water that Paulina drinks throughout the lesson. Notice also that this value is close the estimate found in Part B because 23 is equivalent to 69. ccc Result & & Estimate 5/9 & ≈ & 6/9 Since the actual result and the estimate are close to each other, the answer 59 is reasonable.

Part of 1 512-hour Class
Activity	Fraction
Warm-up	1/5
Instruction	1/2
Playing game	1/10
Cool-down	1/5

The sum of the fractions is 25. 1/5 + 1/5 = 2/5 Next, the time spent on warming up and cooling down will be found by multiplying this fraction number by the lesson time 1 512. This will give the amount of time in hours. 2/5* 1 512 This is a multiplication of a fraction by a mixed number. Before performing the multiplication, the mixed number will be rewritten as an improper fraction. Now multiply the numerators and denominators. Since a common factor exists between numerator and denominator, the fraction can be simplified. 34 /2/60 / 2 = 17/30 The fraction simplifies to 1730. This means that 1730 of an hour is spent on warming up and cooling down. Now the number of minutes spent on these activities will be found by multiplying the fraction by 60 because 1 hour is 60 minutes. 60* 17/30 This is a multiplication of a whole number by an improper fraction. Start by rewriting 60 as an improper fraction, then multiply the fractions. The warm-up and cool-down activities last for 34 minutes.

Part of 1 512-hour Class
Activity	Fraction
Warm-up	1/5
Instruction	1/2
Playing game	1/10
Cool-down	1/5

As shown, half of the class time is spent on instruction. In other words, one-half of the lesson is not spent on instruction. With this in mind, the total lesson time will be multiplied by 12. 1/2 * 1 5/12 Rewrite the mixed number as an improper fraction to find the product. Next, multiply the numerators and denominators. This fraction cannot be simplified further. It means that 1724 of an hour is spent on activities other than instruction. Finally, multiply 1724 by 60 to write the amount of time in terms of minutes. 60* 17/24 This is a multiplication of a whole number by a fraction. An easy way to find this product is to move the whole number to the numerator of the fraction as a factor. The activities other than instruction last for 42.5 minutes.

The length of the photo is 8 34 inches and the width of the photo is 6 23 inches. Area = 8 34 * 6 23 Since the fractional parts of the mixed numbers are greater than 12, the mixed numbers can be rounded up. Area = & 8 34 * 6 23 & ↓ ↓ & 9 * 7 Therefore, the area of the photo is about 9* 7, or 63 square inches.

Area = 8 34 * 6 23 To multiply these mixed numbers, they first need to be rewritten as improper fractions. Recall that the product of two fractions is equal to the product of the numerators divided by the product of the denominators. The area of the photo is 58 13 square inches. Note that this is about the same as the estimate found in Part A, so this answer is reasonable. The expression for the area of the photo with the frame is then the product of the side lengths shown in the diagram. (8 34+2* 1/5) * (6 23+2* 1/5) ⇕ (8 34+2/5) * (6 23+ 2/5) Once again, rewrite the fractions so that all of them are either proper or improper fractions. When multiplying fractions, the product is equal to the product of the numerators divided by the product of the denominators. The area of the photo with the frame is 64 3350 square inches.

It is given that Paulina has 5 hours of free time per day and that two-sixths of this time is devoted to physical activities. The time spent on running can be found by finding two-sixths of 5. 2/6 * 5 Notice that the fraction can be simplified to its lowest terms. Paulina runs for 53 hours every day. Since the number of hours she runs in 4 days is required, the daily amount of activity should be multiplied by 4 to find the total amount of time. This final fraction can also be written as a mixed number. In 4 days, Paulina runs 6 23 hours.