Advanced Decimals: Place Value, Operations, and Applications

1. Word Problem (3 pts)

A jacket marked at $48.75 is sold with a 15% discount, and then an 8% sales tax is applied to the discounted price. What is the final price to the nearest cent?

✎ Worked Solution

Step 1: Discount = 48.75 × 0.15 = 7.3125

Step 2: Discounted price = 48.75 − 7.3125 = 41.4375

Step 3: Tax = 41.4375 × 0.08 = 3.315

Step 4: Final price = 41.4375 + 3.315 = 44.7525

Answer: $44.75

2. Word Problem (3 pts)

A 12.6 m rope is cut into three pieces. Two pieces measure 2.75 m and 4.8 m. The remaining rope is used to make two ties of 0.95 m each. How many meters of rope are left unused?

✎ Worked Solution

Step 1: Sum of first two pieces = 2.75 + 4.8 = 7.55 m

Step 2: Rope after two pieces = 12.6 − 7.55 = 5.05 m

Step 3: Rope used for ties = 2 × 0.95 = 1.90 m

Step 4: Leftover = 5.05 − 1.90 = 3.15 m

Answer: 3.15 m

3. Word Problem (3 pts)

Two bottles contain 1.25 L and 0.85 L of juice. They are poured into a carafe that can hold 1.90 L. How much overflows?

✎ Worked Solution

Step 1: Total juice = 1.25 + 0.85 = 2.10 L

Step 2: Overflow = 2.10 − 1.90 = 0.20 L

Answer: 0.20 L

4. Word Problem (3 pts)

A runner’s lap times (in minutes) are 1.28, 1.31, 1.27, and 1.35. What is the average time, and by how many seconds does it differ from a target average of 1.30 minutes?

✎ Worked Solution

Step 1: Sum = 1.28 + 1.31 + 1.27 + 1.35 = 5.21 min

Step 2: Average = 5.21 ÷ 4 = 1.3025 min

Step 3: Difference from 1.30 = 1.3025 − 1.30 = 0.0025 min

Step 4: Convert to seconds = 0.0025 × 60 = 0.15 s

Answer: Average = 1.3025 min; Difference = 0.15 s above target

5. Word Problem (3 pts)

On a number line, A = 2.45 and B = 7.08. Point P is 1.75 units to the right of A. Find the coordinate of P and the distance from P to B.

✎ Worked Solution

Step 1: P = 2.45 + 1.75 = 4.20

Step 2: Distance PB = 7.08 − 4.20 = 2.88

Answer: P = 4.20; Distance PB = 2.88

6. Word Problem (3 pts)

A measurement rounds to 5.62 when rounded to the nearest hundredth. State the minimum possible value, the maximum possible value (upper bound), and the width of this interval.

✎ Worked Solution

Step 1: For nearest hundredth, bounds are [5.615, 5.625)

Step 2: Minimum = 5.615; Maximum (upper bound) = 5.625 (not included)

Step 3: Width = 5.625 − 5.615 = 0.010

Answer: Min = 5.615; Max < 5.625; Interval width = 0.01

7. Word Problem (3 pts)

A shopper buys 3.5 kg of apples at $2.39 per kg and 1.8 kg of bananas at $1.75 per kg, paying with a $20 bill. Round only the final change to the nearest cent. How much change is received?

✎ Worked Solution

Step 1: Apple cost = 3.5 × 2.39 = 8.365

Step 2: Banana cost = 1.8 × 1.75 = 3.15

Step 3: Total cost = 8.365 + 3.15 = 11.515

Step 4: Change (unrounded) = 20.00 − 11.515 = 8.485

Step 5: Round change to nearest cent = $8.49

Answer: $8.49

8. Word Problem (3 pts)

A cyclist travels at 52.5 km/h for 1.6 hours and then at 48.3 km/h for 0.75 hours. What total distance does the cyclist cover?

✎ Worked Solution

Step 1: Distance 1 = 52.5 × 1.6 = 84.0 km

Step 2: Distance 2 = 48.3 × 0.75 = 36.225 km

Step 3: Total distance = 84.0 + 36.225 = 120.225 km

Answer: 120.225 km

9. Word Problem (3 pts)

A rectangular garden is 8.4 m by 3.75 m. A uniform border of width 0.15 m is added around it. What is the area of the border?

✎ Worked Solution

Step 1: Inner area = 8.4 × 3.75 = 31.5 m²

Step 2: Outer dimensions = (8.4 + 2×0.15) by (3.75 + 2×0.15) = 8.7 by 4.05

Step 3: Outer area = 8.7 × 4.05 = 35.235 m²

Step 4: Border area = 35.235 − 31.5 = 3.735 m²

Answer: 3.735 m²

10. Word Problem (3 pts)

The sum of three decimal numbers is 23.57. Two of them are 8.946 and 7.800. Find the third number.

✎ Worked Solution

Step 1: Sum of known numbers = 8.946 + 7.800 = 16.746

Step 2: Third number = 23.57 − 16.746 = 6.824

Answer: 6.824

11. Word Problem (3 pts)

A solution is made by mixing 0.50 L of 2.4% salt solution and 0.80 L of 1.5% salt solution. Assuming 1 L = 1000 g of solution, how many grams of salt are in the mixture?

✎ Worked Solution

Step 1: Mass in first portion = 0.50 L × 1000 g/L = 500 g; salt = 500 × 0.024 = 12 g

Step 2: Mass in second portion = 0.80 L × 1000 g/L = 800 g; salt = 800 × 0.015 = 12 g

Step 3: Total salt = 12 + 12 = 24 g

Answer: 24 g

12. Word Problem (3 pts)

A car uses 3.6 L of fuel to travel 45.0 km and 2.9 L to travel another 38.0 km. What is the overall fuel consumption in L/100 km (rounded to two decimal places)?

✎ Worked Solution

Step 1: Total fuel = 3.6 + 2.9 = 6.5 L

Step 2: Total distance = 45.0 + 38.0 = 83.0 km

Step 3: L/100 km = (6.5 ÷ 83.0) × 100 = 7.831325...

Step 4: Round to two decimals = 7.83 L/100 km

Answer: 7.83 L/100 km

13. Word Problem (3 pts)

Solve for x: 3(1.2x − 0.45) = 2.7 + 0.9x.

✎ Worked Solution

Step 1: Expand left: 3 × 1.2x = 3.6x; 3 × (−0.45) = −1.35

Step 2: Equation: 3.6x − 1.35 = 2.7 + 0.9x

Step 3: Subtract 0.9x: 2.7x − 1.35 = 2.7

Step 4: Add 1.35: 2.7x = 4.05

Step 5: Divide: x = 4.05 ÷ 2.7 = 1.5

Answer: x = 1.5

14. Word Problem (3 pts)

An arithmetic sequence starts at 0.85 and increases by 0.35 each term. What is the sum of the first 10 terms?

✎ Worked Solution

Step 1: a1 = 0.85, d = 0.35, n = 10

Step 2: Sum formula: S_n = n/2 [2a1 + (n−1)d]

Step 3: S_10 = 10/2 [2(0.85) + 9(0.35)] = 5[1.7 + 3.15] = 5 × 4.85

Step 4: Compute: 5 × 4.85 = 24.25

Answer: 24.25

15. Word Problem (3 pts)

A taxi charges a base fee of $3.50, $1.75 per km for the first 5 km, and $1.20 per km thereafter. How much does a 12.6 km ride cost?

✎ Worked Solution

Step 1: First 5 km cost = 5 × 1.75 = 8.75

Step 2: Remaining distance = 12.6 − 5 = 7.6 km

Step 3: Remaining cost = 7.6 × 1.20 = 9.12

Step 4: Total fare = 3.50 + 8.75 + 9.12 = 21.37

Answer: $21.37

16. Word Problem (3 pts)

A recipe uses 0.75 kg of flour to make 12 cookies. How many kilograms and grams of flour are needed to make 30 cookies?

✎ Worked Solution

Step 1: Scale factor = 30 ÷ 12 = 2.5

Step 2: Flour needed (kg) = 0.75 × 2.5 = 1.875 kg

Step 3: Convert to grams = 1.875 × 1000 = 1875 g

Answer: 1.875 kg (which is 1875 g)

17. Word Problem (3 pts)

A worker logged 7.6 h on Monday, 8.25 h on Tuesday, and 6.9 h on Wednesday. To average at least 8.0 h per day over a 5-day week, how many total hours must be worked on Thursday and Friday combined?

✎ Worked Solution

Step 1: Required total = 5 × 8.0 = 40.0 h

Step 2: Hours so far = 7.6 + 8.25 + 6.9 = 22.75 h

Step 3: Needed Thu+Fri = 40.0 − 22.75 = 17.25 h

Answer: 17.25 h

18. Word Problem (3 pts)

A stock rises from $12.48 to $14.10. What is the percent increase, rounded to two decimal places?

✎ Worked Solution

Step 1: Increase = 14.10 − 12.48 = 1.62

Step 2: Percent increase = (1.62 ÷ 12.48) × 100 = 12.980769...

Step 3: Round to two decimals = 12.98%

Answer: 12.98%

19. Word Problem (3 pts)

A bucket holds 12.5 L of water. A leak drips at a constant rate of 0.075 L per minute into the bucket. How long will it take to collect 3/5 of the bucket’s capacity?

✎ Worked Solution

Step 1: Target volume = (3/5) × 12.5 = 7.5 L

Step 2: Time = 7.5 ÷ 0.075 = 100 minutes

Answer: 100 minutes

20. Word Problem (3 pts)

Two phone plans are compared for 42 calls: Plan A charges $19.95 per month plus $0.35 per call; Plan B charges a flat $31.50. Which plan is cheaper and by how much?

✎ Worked Solution

Step 1: Plan A variable = 0.35 × 42 = 14.70

Step 2: Plan A total = 19.95 + 14.70 = 34.65

Step 3: Plan B total = 31.50

Step 4: Difference = 34.65 − 31.50 = 3.15

Answer: Plan B is cheaper by $3.15

21. Fill in the Blank (3 pts)

Compute (4.2 + 0.86) − 1.375 = _____

✓ Answer

3.685

22. Fill in the Blank (3 pts)

7.5 ÷ 0.12, rounded to the nearest tenth, equals _____

✓ Answer

62.5

23. Fill in the Blank (3 pts)

The digit d that makes 7.d89 round to 7.9 (nearest tenth) and 7.89 (nearest hundredth) is _____

✓ Answer

8

24. Fill in the Blank (3 pts)

Solve for x: 0.4x + 1.27 = 3.03 ⇒ x = _____

✓ Answer

4.4

25. Fill in the Blank (3 pts)

The combined mass of 0.845 kg, 0.915 kg, and 1.075 kg is _____ kg

✓ Answer

2.835

26. Fill in the Blank (3 pts)

Write 17/40 as a decimal: _____

✓ Answer

0.425

27. Fill in the Blank (3 pts)

Complete: 2.4 + _____ = 3(1.05)

✓ Answer

0.75

28. Fill in the Blank (3 pts)

Evaluate 0.048 × 125 = _____

✓ Answer

6

29. Fill in the Blank (3 pts)

If y = 1.2x and x = 0.35, then y = _____

✓ Answer

0.42

30. Fill in the Blank (3 pts)

The mean of 5.84, 6.12, and 6.74 rounded to the nearest hundredth is _____

✓ Answer

6.23

31. Fill in the Blank (3 pts)

The number that satisfies 0.25 × _____ = 3.75 is _____

✓ Answer

15

32. Fill in the Blank (3 pts)

Round 8.945 to the nearest hundredth: _____

✓ Answer

8.95

33. Fill in the Blank (3 pts)

Compute 9.8 ÷ 0.02 = _____

✓ Answer

490

34. Fill in the Blank (3 pts)

With exactly three decimal places, the least number that rounds to 2.31 (nearest hundredth) is _____

✓ Answer

2.305

35. Fill in the Blank (3 pts)

Find t: 6.4 + t = 2.5 × 3.2 ⇒ t = _____

✓ Answer

1.6

36. Fill in the Blank (3 pts)

Convert 3 km 45 m to kilometers as a decimal: _____ km

✓ Answer

3.045

37. Fill in the Blank (3 pts)

A rectangle’s area is 12.96 m² and one side is 3.2 m. The other side is _____ m

✓ Answer

4.05

38. Fill in the Blank (3 pts)

Decrease 5.2 by 15%: _____

✓ Answer

4.42

39. Fill in the Blank (3 pts)

Let p = 0.36 × 2.5. Then p + 1.04 = _____

✓ Answer

1.94

40. Fill in the Blank (3 pts)

Complete the equation: 7.203 + _____ = 10.5

✓ Answer

3.297