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Question 181Question →

A municipal election committee is distributing ballots to various polling stations. When they pack the ballots in bundles of $45$ , $54$ , or $72$ , they find that they are always left with $38$ , $47$ , and $65$ unbundled ballots, respectively. If the total number of ballots printed is the largest possible 4-digit number satisfying these conditions, what is the exact total number of ballots?

Show answer & explanation

Answer: 9713

Answer

9713

The problem describes a scenario where the difference between each divisor (

45

,

54

,

72

) and its respective remainder (

38

,

47

,

65

) is exactly

7

. This means that if

7

more ballots were added, the total would be perfectly divisible by all three numbers. Therefore, the required total is exactly

7

less than a common multiple of these divisors. The LCM of

45

,

54

, and

72

is

1080

. The largest

4

-digit multiple of

1080

is

9720

(

1080 \times 9

). Subtracting the constant difference of

7

from

9720

gives the final answer of

9713

.

Step-by-Step Solution

1

Calculate the difference between each bundle size and its corresponding remainder.

45 - 38 = 7

,

54 - 47 = 7

, and

72 - 65 = 7

.

To identify if there is a constant difference, which allows the use of the LCM minus constant method.

2

Determine the Least Common Multiple (LCM) of the bundle sizes

45

,

54

, and

72

.

\text{LCM}(45, 54, 72) = 1080

.

The LCM represents the smallest bundle size that perfectly divides by all three numbers. Prime factorizations:

45 = 3^2 \times 5

,

54 = 2 \times 3^3

,

72 = 2^3 \times 3^2

. LCM

= 2^3 \times 3^3 \times 5 = 1080

.

3

Find the largest 4-digit multiple of the LCM.

1080 \times 9 = 9720

.

The problem asks for the largest 4-digit number. Dividing

9999

by

1080

yields

9.258...

, so the largest integer multiplier is

9

.

4

Subtract the constant difference from this largest multiple.

9720 - 7 = 9713

.

Since each division left a remainder that was

7

short of a full bundle, subtracting

7

from a perfect multiple satisfies all three remainder conditions.

Key Concept

Solving simultaneous remainder problems where the difference between divisors and remainders is constant, by utilizing the LCM and scaling to a specific boundary range.

Question 182Question →

Consider the following set of 8 numbers:

S = \left\{ -7, 0, \pi, \frac{22}{7}, \sqrt{169}, \sqrt{50}, 2.5, 0.\overline{45} \right\}

Let:
$P =$ the number of integers in $S$
$Q =$ the number of rational numbers in $S$
$R =$ the number of irrational numbers in $S$
$T =$ the number of whole numbers in $S$

Calculate the exact value of $(Q \times R) + (P \times T)$ .

Show answer & explanation

Answer: 18

Answer

18

Based on mathematical definitions: P (integers) = 3 because -7, 0, and 13 are integers. Q (rationals) = 6 because it includes the 3 integers plus 22/7, 2.5, and 0.4545... R (irrationals) = 2 because it includes pi and sqrt(50). T (whole numbers) = 2 because it includes the non-negative integers 0 and 13. Plugging these into (Q * R) + (P * T) yields (6 * 2) + (3 * 2) = 18.

Step-by-Step Solution

1

Evaluate and simplify each number in the set to determine its properties.

\sqrt{169} simplifies to exactly 13. The repeating decimal 0.\overline{45} can be written as a fraction (45/99).

Numbers must be evaluated in their simplest form to avoid misclassification.

2

Determine the value of P by counting the integers.

The integers are -7, 0, and 13. Therefore, P = 3.

Integers include positive numbers, negative numbers, and zero, with no decimal or fractional parts.

3

Determine the value of Q by counting the rational numbers.

The rational numbers are -7, 0, 22/7, 13, 2.5, and 0.\overline{45}. Therefore, Q = 6.

Rational numbers are any numbers that can be expressed as a ratio of two integers.

4

Determine the value of R by counting the irrational numbers.

The irrational numbers are \pi and \sqrt{50}. Therefore, R = 2.

Irrational numbers have non-terminating, non-repeating decimal expansions.

5

Determine the value of T by counting the whole numbers.

The whole numbers are 0 and 13. Therefore, T = 2.

Whole numbers consist only of the non-negative integers.

6

Substitute the counts into the requested mathematical expression and calculate the final result.

(6 \times 2) + (3 \times 2) = 12 + 6 = 18.

To answer the specific question given in the stem.

Key Concept

Classification of real numbers into integers, rational numbers, irrational numbers, and whole numbers.

Question 183Question →

A jeweler is cutting equal-length pieces of gold wire from three different spools to make uniform necklace links without any wastage. The three spools contain gold wire of lengths $\frac{27}{2}$ cm, $\frac{45}{4}$ cm, and $\frac{63}{5}$ cm. What is the maximum possible length of each uniform gold wire piece that can be cut? (Provide your answer as a precise decimal)

Show answer & explanation

Answer: 0.45

Answer

0.45

The maximum uniform length is found by computing the Highest Common Factor (HCF) of the fractional wire lengths. Applying the formula yields an HCF of numerators (9) over the LCM of denominators (20), giving

\frac{9}{20}

, which correctly converts to exactly

0.45

cm.

Step-by-Step Solution

1

Determine the mathematical operation required for the scenario.

Calculate the Highest Common Factor (HCF) of the three fractional lengths.

Cutting multiple lengths into the maximum possible equal segments without wastage is defined by the HCF.

2

Apply the rule for finding the HCF of fractions.

HCF = HCF(Numerators) / LCM(Denominators)

This is the standard formula for finding the greatest common divisor of multiple fractional values.

3

Find the HCF of the numerators 27, 45, and 63.

HCF(27, 45, 63) = 9

9 is the largest integer that divides perfectly into 27, 45, and 63.

4

Find the LCM of the denominators 2, 4, and 5.

LCM(2, 4, 5) = 20

20 is the smallest integer that is a multiple of 2, 4, and 5.

5

Combine the results into the final fraction and convert to a decimal.

\frac{9}{20} = 0.45

cm

The question requires the precise decimal representation of the fraction.

Key Concept

Calculating the Highest Common Factor (HCF) of fractions to solve optimization word problems.

Question 184Question →

A botanical garden is installing a new irrigation system and has three main supply hoses measuring $\frac{44}{5}$ meters, $\frac{77}{4}$ meters, and $\frac{121}{10}$ meters in length. The landscaping team needs to cut all three hoses into smaller, equal-length segments to connect to individual planters. If no material can be wasted, what is the minimum total number of segments that can be produced from all three hoses combined?

Show answer & explanation

Answer: 73

Answer

73 segments

The minimum number of total segments is achieved when each segment is cut to its maximum possible equal length. This maximum length is the HCF of the three fractions (44/5, 77/4, 121/10), which evaluates to 11/20 meters. Dividing the original lengths by this HCF yields exactly 16, 35, and 22 pieces respectively, totaling 73 individual segments.

Step-by-Step Solution

1

Determine the mathematical requirement for minimizing the number of segments.

Identify that the Highest Common Factor (HCF) of the hose lengths must be found.

To get the minimum number of pieces, each piece must be of the maximum possible equal length.

2

Apply the rule for finding the HCF of fractions.

Use the formula:

\text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}

.

This formula allows the calculation of the greatest common divisor for non-integer fractional values.

3

Calculate the HCF of the numerators.

The numerators are

44

,

77

, and

121

. Their HCF is

11

.

The largest integer that perfectly divides

44

,

77

, and

121

is

11

(

44 = 11 \times 4

,

77 = 11 \times 7

,

121 = 11 \times 11

).

4

Calculate the LCM of the denominators.

The denominators are

5

,

4

, and

10

. Their LCM is

20

.

The smallest integer that is a multiple of

5

,

4

, and

10

is

20

.

5

Determine the maximum segment length.

The segment length is

\frac{11}{20}

meters.

Combining the results from the previous steps using the fraction HCF formula yields the length.

6

Calculate the number of segments produced from each hose.

First hose:

\frac{44}{5} \div \frac{11}{20} = 16

. Second hose:

\frac{77}{4} \div \frac{11}{20} = 35

. Third hose:

\frac{121}{10} \div \frac{11}{20} = 22

.

Dividing the total length of each hose by the length of one segment gives the segment count per hose.

7

Sum the segment counts.

16 + 35 + 22 = 73

segments.

The question asks for the minimum total number of segments produced from all three hoses combined.

Key Concept

Calculating the Highest Common Factor (HCF) of fractions and applying it to optimize division in real-world scenarios.

Estimated Time:2m 0s

Question 185Question →

A teacher provides a list of eight numerical values on the board for a classification exercise:

(I) $\sqrt{144}$
(II) $\frac{22}{7}$
(III) $\pi$
(IV) $0.\overline{36}$
(V) $\sqrt{12}$
(VI) $\frac{\sqrt{3}}{\sqrt{27}}$
(VII) $3.14159$
(VIII) $0.121221222...$ (where the number of 2s increases by one each time)

What is the exact count of rational numbers in this list?

Show answer & explanation

Answer: 5

Answer

There are exactly 5 rational numbers in the provided list.

By simplifying each expression and applying the definitions of real numbers, exactly 5 of the 8 values (values I, II, IV, VI, and VII) satisfy the conditions of being a rational number.

Step-by-Step Solution

1

Define the criteria for a rational number.

A rational number is any number that can be expressed as a fraction of two integers (p/q, where q is not zero). This includes all integers, terminating decimals, and repeating decimals.

Establishing the definition is necessary to accurately classify each item.

2

Evaluate the square roots and fractions in the list.

\sqrt{144} = 12

, which is an integer (Rational).

\frac{22}{7}

is a ratio of two integers (Rational).

\sqrt{12} = 2\sqrt{3}

, which contains the root of a non-perfect square (Irrational).

\frac{\sqrt{3}}{\sqrt{27}}

simplifies to

\sqrt{\frac{3}{27}} = \sqrt{\frac{1}{9}} = \frac{1}{3}

(Rational).

Radicals must be simplified to their lowest terms to reveal their true mathematical classification.

3

Evaluate the decimal representations and constants in the list.

0.\overline{36}

is a repeating decimal (Rational).

3.14159

is a terminating decimal (Rational).

\pi

is a transcendental constant with infinite, non-repeating digits (Irrational).

0.121221222...

has a changing pattern, making it non-terminating and non-recurring (Irrational).

Decimals must be analyzed by their termination or repetition properties.

4

Count the total number of items identified as rational.

The rational values are (I), (II), (IV), (VI), and (VII). The total count is 5.

The question asks for the exact numerical count of rational numbers.

Key Concept

Identifying rational and irrational numbers by their fractional, radical, and decimal properties.

Question 186Question →

An urban traffic control system manages three independent electronic toll gates. Based on their internal sensor loops, Gate A completes its automated scanning cycle every $\frac{12}{5}$ seconds, Gate B every $\frac{18}{25}$ seconds, and Gate C every $\frac{27}{35}$ seconds. If all three gates reset their cycles simultaneously, how many seconds will it take for all three gates to reset simultaneously again? Express your answer as an exact decimal.

Show answer & explanation

Answer: 21.6

Answer

21.6

Simultaneous repeating events require finding the Lowest Common Multiple (LCM) of their individual periods. For fractional periods, the rule is

\text{LCM} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}

. The LCM of

12

,

18

, and

27

is

108

, and the HCF of

5

,

25

, and

35

is

5

. Therefore, the LCM of the fractions is

\frac{108}{5}

, which evaluates precisely to

21.6

.

Step-by-Step Solution

1

Identify the mathematical operation required.

Calculate the LCM of the fractions

\frac{12}{5}

,

\frac{18}{25}

, and

\frac{27}{35}

.

Simultaneous cyclic events coincide at the Lowest Common Multiple of their individual time intervals.

2

Determine the LCM of the numerators.

The LCM of

12

,

18

, and

27

is

108

.

The formula for the LCM of fractions requires calculating the LCM of their respective numerators.

3

Determine the HCF of the denominators.

The HCF of

5

,

25

, and

35

is

5

.

The formula for the LCM of fractions requires calculating the HCF of their respective denominators.

4

Apply the fraction LCM formula.

\frac{108}{5} = 21.6

Dividing the LCM of numerators by the HCF of denominators yields the final LCM of the given fractions.

Key Concept

Lowest Common Multiple (LCM) of Fractions

Question 187Question →

A specialized aerospace component is manufactured from a custom metal alloy weighing exactly $160$ kg. The alloy's composition by weight is $0.35$ aluminum and $\frac{5}{16}$ magnesium, with the remaining portion consisting entirely of titanium. What is the exact mass of titanium in this component?

Show answer & explanation

Answer: 54

Answer

54

To find the mass of titanium, the total proportions of the other metals must first be unified into a common format (either fractions or decimals). By converting the magnesium's share of

\frac{5}{16}

to the decimal

0.3125

, the combined proportion of aluminum and magnesium is

0.35 + 0.3125 = 0.6625

. The titanium makes up the remaining portion of the whole, which is

1 - 0.6625 = 0.3375

. Finally, multiplying this decimal proportion by the total alloy mass of

160

kg yields exactly

54

kg.

Step-by-Step Solution

1

Convert the fractional part to a decimal.

\frac{5}{16} = 0.3125

Unifying the terms into a single format (decimals) makes addition straightforward.

2

Add the portions of aluminum and magnesium.

0.35 + 0.3125 = 0.6625

This finds the total proportion of the alloy that is NOT titanium.

3

Calculate the proportion of titanium.

1 - 0.6625 = 0.3375

The sum of all material proportions in the whole alloy must equal exactly

1

.

4

Calculate the final mass of titanium.

0.3375 \times 160 = 54

Multiplying the component's proportion by the total mass gives the specific weight of that component.

Key Concept

Converting between fractions and decimals and applying proportional reasoning to find a part of a whole.

Question 188Question →

A civil contractor is tasked with upgrading a rural highway. In the first month, the crew successfully upgrades $0.275$ of the total highway length. During the second month, they upgrade $\frac{5}{12}$ of the remaining length. If the crew has exactly $10.15$ kilometers left to upgrade in the third month to complete the project, what is the total length of the highway in kilometers?

Show answer & explanation

Answer: 24

Answer

The total length of the highway is 24 kilometers.

The correct total length is calculated by accurately determining the remaining fraction of the highway step-by-step. Converting the decimal 0.275 to a fraction (11/40) allows for clean, exact operations without repeating decimal rounding errors. By tracking the remainder accurately, we establish that the final 10.15 km represents exactly 203/480 of the total length, resulting in a total of 24 km.

Step-by-Step Solution

1

Determine the fraction of the highway remaining after the first month.

\frac{29}{40}

of the total length remains.

The crew upgraded

0.275

of the total. Converting

0.275

to a fraction gives

\frac{275}{1000}

, which simplifies to

\frac{11}{40}

. Subtracting this from the whole gives

1 - \frac{11}{40} = \frac{29}{40}

.

2

Calculate the fraction of the total highway upgraded in the second month.

\frac{29}{96}

of the total length.

The crew upgraded

\frac{5}{12}

of the remaining length. Therefore, multiply the two fractions:

\frac{5}{12} \times \frac{29}{40} = \frac{1 \times 29}{12 \times 8} = \frac{29}{96}

.

3

Calculate the total fraction of the highway remaining for the third month.

\frac{203}{480}

of the total length.

Subtract the second month's progress from the remainder after the first month:

\frac{29}{40} - \frac{29}{96}

. The least common multiple of

40

and

96

is

480

. Converting to common denominators gives

\frac{348}{480} - \frac{145}{480} = \frac{203}{480}

.

4

Set up the final equation and solve for the total highway length.

24

km

The remaining fraction equals the given physical distance of

10.15

km. So,

\frac{203}{480} \times \text{Total} = 10.15

. Solving for Total yields

\frac{10.15 \times 480}{203}

. Since

10.15 \div 203 = 0.05

, the Total is

0.05 \times 480 = 24

km.

Key Concept

Solving sequential parts-of-a-whole word problems by effectively converting between decimals and fractions to find a remaining proportion.

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