Mathematics Proportions and Ratios Quiz
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Questions and Answers

What is the weight of the nitrogen compound in 20 kg of fertilizer?

  • 3 kg
  • 3.6 kg (correct)
  • 2 kg
  • 4 kg

The weight of the potassium compound is 3.6 kg.

False (B)

How much petrol does the car consume to cover 200 km?

20 litres

The bullock-cart takes __________ hours to cover a distance of 100 km at the speed of 5 km/hr.

<p>20</p> Signup and view all the answers

Match the following quantities with their types of proportion:

<p>Nitrogen compound weight in fertilizer = Direct proportion Speed and time of the Car = Inverse proportion Distance covered per litre of petrol = Direct proportion Speed and time of the Bullock-cart = Inverse proportion</p> Signup and view all the answers

Which of the following statements best describes direct variation?

<p>As one quantity increases, the other also increases proportionally. (B)</p> Signup and view all the answers

The ratio of speed and time for the bullock-cart is a direct variation.

<p>False (B)</p> Signup and view all the answers

In a direct variation, the product of two quantities is __________.

<p>constant</p> Signup and view all the answers

What method uses the equation $\frac{a}{b} = k$ to relate variables?

<p>Invertendo (C)</p> Signup and view all the answers

The Componendo method states that if $\frac{a}{b} = \frac{c}{d}$ then $a + b = c + d$.

<p>False (B)</p> Signup and view all the answers

If $ba = 74$, what is the next step to find the ratio $5ab - b$?

<p>Substitute $a$ from $a = \frac{74}{b}$ and then calculate $5(\frac{74}{b})b - b$.</p> Signup and view all the answers

The method that relates $\frac{ba}{dc}$ to $\frac{a+b}{c+d}$ is called __________.

<p>Componendo</p> Signup and view all the answers

Match the following methods with their definitions:

<p>Invertendo = Switches numerator and denominator Componendo = Combines numerators and denominators Alternando = Expresses variable ratios in a different form Dividendo = Relates to dividing terms</p> Signup and view all the answers

If $\frac{b}{d} = 3$, which of the following indicates the relationship between $a$ and $b$ using the Alternando method?

<p>$a = \frac{b}{3}$ (C)</p> Signup and view all the answers

The relationship $\frac{a+c}{b+d}$ holds true in all cases of the Componendo method.

<p>False (B)</p> Signup and view all the answers

Using the Componendo-Dividendo method, if $ba = dc$, express $\frac{a-b}{c-d}$.

<p>The ratio is $\frac{a - b}{c - d} = \frac{b(c-d)}{d(a-b)}$.</p> Signup and view all the answers

What are the values of x and y that solve the equations x + y = 14 and x - y = 2?

<p>(8, 6) (B)</p> Signup and view all the answers

The general form of a linear equation in two variables is ax + by + c = 0, where both a and b can be zero.

<p>False (B)</p> Signup and view all the answers

If the sum of the ages of the mother and son is 45 years, and the mother's age is x, what is the son's age in terms of x?

<p>45 - x</p> Signup and view all the answers

The simultaneous equations can be solved by eliminating one of the __________.

<p>variables</p> Signup and view all the answers

Match the following equations with their respective solutions:

<p>x + y = 14 = (8, 6) x - y = 2 = (8, 6) 2x + 3y = 1 = (−1, 1) 3x - 4y - 15 = 0 = (5, 0)</p> Signup and view all the answers

What is the age of the son if the mother's age is 33?

<p>12 (B)</p> Signup and view all the answers

The equation 3x + y - 5 = 0 is in the standard form of a linear equation.

<p>True (A)</p> Signup and view all the answers

What is the final value of y when solving the equations x + y = 45 and 2x - y = 54?

<p>12</p> Signup and view all the answers

What is the value of x in the equation 3x = 16 + 4y when y = 2?

<p>8 (C)</p> Signup and view all the answers

The solution to the equation 3x - 4y = 16 is (8, 2).

<p>True (A)</p> Signup and view all the answers

What is the value of y when x is 1 in the equation y = 3x - 2?

<p>1</p> Signup and view all the answers

The coordinates of the solution of the given equations are (__, __).

<p>(8, 2)</p> Signup and view all the answers

Match the equations with their corresponding transformations:

<p>3x = 16 + 4y = y = 3x - 2 3x - y = 2 = 17x - 6 = 11 2x - 3y = 10 = 3x - 4y = 16 8x + 3y = 11 = x = 8</p> Signup and view all the answers

What do you get when you solve the equation 2(16 + 4y)/3 - 3y = 10?

<p>y = 2 (B)</p> Signup and view all the answers

The equation 8x + 9x - 6 = 11 simplifies to x = 7.

<p>False (B)</p> Signup and view all the answers

Write one solution of the equation x + y = 7.

<p>(3, 4)</p> Signup and view all the answers

What is the total planned expenditure for Mr. Shah?

<p>Rs. 1,58,000 (B)</p> Signup and view all the answers

Mr. Shah has Rs. 4,82,000 available for yearly expenses after planned expenditures.

<p>True (A)</p> Signup and view all the answers

How much interest did Mr. Shah earn from his bank investment?

<p>Rs. 67,200</p> Signup and view all the answers

Mr. Shah invested Rs. _________ in mutual funds.

<p>2,40,000</p> Signup and view all the answers

Match the following investments to their outcomes:

<p>Bank Investment = Rs. 67,200 interest Mutual Funds = Rs. 3,05,000 after 2 years Insurance Premium = Rs. 24,000 annually Provident Fund Contribution = Rs. 1,28,000 annually</p> Signup and view all the answers

Which of Mr. Shah's investments yielded a higher profit?

<p>Mutual Funds (D)</p> Signup and view all the answers

Mr. Shah's total annual income is less than Rs. 6,40,000.

<p>False (B)</p> Signup and view all the answers

What formula is used to calculate compound interest?

<p>I = A - P</p> Signup and view all the answers

What is the total cost of 8 books and 5 pens?

<p>420 rupees (A)</p> Signup and view all the answers

The ratio of incomes of the two persons is 9:7.

<p>True (A)</p> Signup and view all the answers

What is the area change if the length of a rectangle is reduced by 5 units and the breadth is increased by 3 units?

<p>-8 square units</p> Signup and view all the answers

The distance between places A and B is _____ kilometers.

<p>70</p> Signup and view all the answers

Match the financial terms with their definitions:

<p>Savings = Money set aside for future use Investments = Assets purchased to generate income Income Tax = Tax imposed on individual earnings Financial Planning = Process of managing finances for future goals</p> Signup and view all the answers

What happens to the fraction if the numerator is multiplied by 3 and 3 is subtracted from the denominator?

<p>It becomes 18 (C)</p> Signup and view all the answers

If both cars travel towards each other, they meet after 1 hour.

<p>True (A)</p> Signup and view all the answers

How much do both persons save?

<p>200 rupees each</p> Signup and view all the answers

Flashcards

Direct Proportion

A relationship between two quantities where, as one quantity increases, the other quantity also increases at a constant rate.

Inverse Proportion

A relationship between two quantities where, as one quantity increases, the other quantity decreases at a constant rate.

Constant of Proportionality

The ratio between two quantities that are in direct proportion is constant.

Constant of Inverse Proportionality

The product of two quantities that are in inverse proportion is constant.

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Proportion Method

A method to solve problems involving proportion by setting up a ratio and finding the unknown value.

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Inverse Proportion Method

A method to solve problems involving inverse proportion where you find the product of corresponding values to determine the unknown value.

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Proportion

A mathematical relationship where two quantities change in a way that their ratio remains constant.

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Inverse Proportion

A mathematical relationship where two quantities change in a way that their product remains constant.

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Invertendo

A mathematical rule that states that if two ratios are equal, then their reciprocals are also equal.

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Componendo

If two ratios are equal, then the sum of the numerator and denominator of the first ratio divided by the denominator is equal to the sum of the numerator and denominator of the second ratio divided by the denominator.

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Alternando

A mathematical rule that states if two ratios are equal, then the product of the numerator of the first ratio and the denominator of the second ratio is equal to the product of the denominator of the first ratio and the numerator of the second ratio.

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Dividendo

A mathematical rule that states if two ratios are equal, then the difference between the numerator and denominator of the first ratio divided by the denominator is equal to the difference between the numerator and denominator of the second ratio divided by the denominator.

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Componendo-Dividendo

A mathematical rule that combines Componendo and Dividendo. It states that if two ratios are equal, then the sum of the numerator and denominator of the first ratio divided by the difference between the numerator and denominator of the first ratio is equal to the sum of the numerator and denominator of the second ratio divided by the difference between the numerator and denominator of the second ratio.

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Solving Ratios and Proportions

Solving a problem using the rules of ratios and proportions. For example, you may use Componendo, Dividendo, or Alternando rules to simplify or manipulate the given equations.

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Ratio

A numerical representation of the relationship between two quantities. Ratios express how many times one quantity is contained within another quantity.

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Linear Equation in Two Variables

A mathematical equation where the highest power of the variables is 1. It can be represented as ax + by + c = 0, where a, b, and c are real numbers, and both a and b are not zero.

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Simultaneous Equations

A set of two or more equations with the same variables. The goal is to find values that satisfy all the equations simultaneously.

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Elimination Method

A method to solve simultaneous equations by adding or subtracting the equations to eliminate one variable.

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Solution of Simultaneous Equations

A pair of values (x, y) that makes both equations in a system of simultaneous equations true.

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Substitution Method

A way to solve simultaneous equations by isolating one variable in one equation and substituting it into the other equation.

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Coefficient in Linear Equation

In a linear equation, the coefficient of a variable represents the rate of change of the dependent variable with respect to that variable.

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General Form of Linear Equation

The general form of a linear equation in two variables is ax + by + c = 0, where 'a', 'b', and 'c' are real numbers, and 'a' and 'b' are not zero.

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Verification of Solutions

The process of checking whether a proposed solution satisfies all the equations in a system of simultaneous equations.

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Linear equation

An equation where the highest power of the variables is one. This means that the variables are not multiplied by themselves or other variables, only by numbers.

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Ordered pair

A set of two numbers representing the values of the variables (x and y) that satisfy both equations in the system.

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Substituting a variable

The process of replacing a variable in an equation with its equivalent expression from another equation. This simplifies the equation and allows you to solve for the remaining unknown variable.

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Expressing one variable in terms of the other

An equation in which the value of one variable is expressed in terms of the other. This allows for direct substitution of one variable in another equation.

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Total Planned Expenditure

The sum of all money that is spent on things like insurance, savings, and emergencies.

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Amount Available for Yearly Expenses

The amount of money left over after total planned expenditures are deducted from annual income.

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Profit

The gain or profit made on an investment.

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Compound Interest

A method of calculating interest where interest earned is added to the principal amount, and interest is then calculated on the new principal amount.

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Calculating Interest

The process of calculating the interest gained on an investment over a specific period.

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Amount

The total amount of money received after an investment has grown due to interest.

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Principal

The initial sum of money invested.

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Mutual Funds

Funds that are invested in a wide range of securities, aiming to achieve a balanced portfolio.

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Financial planning

The process of planning and managing your finances to achieve your long-term financial goals.

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Income statement

A financial statement that shows the income and expenses of a person or business over a period of time.

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Savings

The amount of money that is set aside for future use.

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Investments

The process of using savings to purchase assets that are expected to increase in value over time.

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Tax structure

A system of rules that determines how much tax an individual or business is required to pay.

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Computation of Income tax

The process of calculating the amount of income tax that an individual or business is required to pay.

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The importance of tax structure

A system of rules that determines how much tax an individual or business is required to pay. The tax structure affects both the income of individuals and businesses as it determines the amount of tax that should be paid.

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The role of income tax computation in financial planning

The process of calculating the amount of income tax that an individual or business is required to pay. The computation of income tax requires careful attention to detail and adherence to the relevant tax laws and regulations.

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Study Notes

Ratio and Proportion

  • Ratio compares two quantities by division. A ratio is written as a:b or a/b.
  • Proportion is an equality of two ratios.
  • Properties of ratios:
    • The ratio a:b can be expressed as ka:kb, where k is a non-zero constant.
    • The ratio a/b is unchanged if both a and b are multiplied or divided by the same non-zero value.
  • Theorem of Equal Ratios: If a/b = c/d = e/f, then (a+c+e)/(b+d+f) = a/b.
  • The k-method: If a/b = c/d = e/f = k, then a = bk, c = dk, e = fk.
  • Direct proportion. Two quantities are directly proportional if an increase in one causes a proportional increase in the other, and vice-versa.
  • Inverse proportion. Two quantities are inversely proportional if an increase in one causes a proportional decrease in the other, and vice-versa.

Properties of Ratio

  • Ratio of two numbers a and b is a:b or a/b, where 'a' is the first term, and 'b' is the second term.
  • In the ratio a/b, if b = 100, then it is a percentage. e.g., 2/100 = 2%
  • The ratio remains unchanged if the terms are multiplied or divided by the same non-zero value. e.g., 3 : 4 = 6:8 = 9:12
  • The quantities taken in the ratio must be in the same units.
  • The ratio of two quantities is dimensionless.
  • If a/b = c/d, then ad = bc.

Comparison of Ratios

  • The numbers a, b, c, d being positive, comparison of ratios a/b and c/d can be done using the following rules:
    • If ad > bc, then a/b > c/d
    • If ad < bc, then a/b < c/d
    • If ad = bc, then a/b = c/d

Operations on Equal Ratios

  • Invertendo: If a/b = c/d, then b/a = d/c
  • Alternando: If a/b = c/d, then a/c = b/d
  • Componendo: If a/b = c/d, then (a + b)/b = (c + d)/d
  • Dividendo: If a/b = c/d, then (a - b)/b = (c - d)/d
  • Componendo-dividendo: If a/b = c/d, then (a + b)/(a - b) = (c + d)/(c - d)

Continued Proportion

  • If a, b, and c are in continued proportion, then a/b = b/c = k.
  • In this case, b is the geometric mean of a and c and b² = ac.

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Ratio and Proportion PDF

Description

This quiz tests your knowledge on proportions, ratios, and variations in mathematics. It includes questions about direct variation, methods of relating variables, and practical applications such as calculating speed, weight, and ratios. Perfect for students looking to strengthen their understanding of mathematical relationships.

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