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mathematics UK Year 10 Mathematics (KS4)

Solved problems for the chapter:

Substituting numbers into expressions and formulas.

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Fractions for KS4

1. Special fractions

2. Equal fraction and additive inverse of a fraction

2.1. Equal (equivalent) fraction

2.1.1. Rational numbers

2.1.2. Fractions greater than 1

2.2. Additive Inverse of a Fraction

3. Order of rational numbers

3.1. Properties of the order relation

3.2. Illustration on a number line

3.3. Signs on fractions

4. Forum

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Absolute and relative error for KS4

1. Mathematical notation

2. Errors

2.1. Absolute error

2.1.1. Absolute error of sum or difference

2.2. Relative error

2.2.1. Relative error of product or quotient

3. Rounding off

3.2.1. Rounding Types

4. Forum

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Quadratic equation

1. General form of the quadratic equation

1.1. Simplifying quadratic equations

1.1.1. Example of simplifying a quadratic equation

1.1.2. Generalisation of simplifying quadratic equation

2. Solving quadratic equations

2.1. Case D> 0

2.2. Case D = 0

2.3. Case D < 0

2.4. Solving special forms of quadratic equations by factorisation

2.4.1. Case b = 0 (generalisation)

2.4.2. Case b = 0 (practical example)

2.4.3. Case c = 0 (generalisation)

2.4.4. Case c = 0 (example)

2.4.5. Case a = 1 (generalisation)

2.4.6. Case a = 1 (example)

3. Zero form of the quadratic equation

4. Forum

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Multiples and expressions for KS4

1. Monomials, binomials and multinomials

2. Breaking down numbers

2.1. Breaking down numbers into prime factors

3. Calculating expressions

3.1. The square of the sum of two terms

3.2. The square of the difference of two terms

3.3. The cube of a binomial

3.4. The difference of squares

3.5. The difference of cubes

3.6. The sum of cubes

3.7. Difference of n-th power

3.8. Vieta's formula

4. Exposing common factors

5. Forum

↓ Show theory [4] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:substituting_numbers_into_expressions_and_formulas/5312/

Example 1.

For the algebraic expression , find its numeric value for which .

Example 2.

Simplify with the use of a calculator and leave your result in for decimal places given that:

For the algebraic expression , find its numeric value for which .

Simplify , given that:

and

For , calculate the value of the expression with a calculator

and round to two significant figures.

For what digits a will the given fraction represent an integer?

Using the formula

where is the gravitational constant, calculate the gravitational acceleration on Mars exactly to three significant figures if the radius is and its mass is .

For what vale of digits a will the given fraction represent an integer, if a represent the last the digit of a three-digit number?

10.

For what digits a will the given fraction represent an integer?

11.

Simplify the expressions below and find its value when :

12.

Given the formula below:

We know that the given formula can be written in the form:

Calculate the value of the expression for

13.

Show whether:

for

14.

Simplify the following expressions and then calculate its value for , and .

15.

Simplify and evaluate, when :

16.

Simplify the expression:

Then determine if the value of the expression is greater when or .

17.

Calculate the value of the expression when:

18.

Calculate the value of the expression when:

19.

The expression is given.

Calculate the value of the expression for .
Calculate the value of the expression for .
Calculate the difference between the values calculated in (a) and (b) above.

20.

A grenade flies out of the cannon at a speed of at an angle relative to the horizontal ground. The time t of the grenade is described by the equation , where the gravitational acceleration on the Earth's surface is .

a) What is the maximum height reached by a grenade? In what time does the grenade reach its maximum height?

b) How long do we have to set the grenade to explode at a height of five meters?

21.

By how much does the value of the expression change if instead of , we insert .

22.

The body that is lowered from the tower to fall freely falls to the ground in 3 seconds. How high is the tower and at what speed does the body fall to the ground? How many paths has the body taken in half the time of falling if we ignore the air resistance?

Use equations , and

23.

Calculate the value of each of the following expressions with the specified values:

For , calculate: .
For , calculate: .

24.

We shoot an arrow at an angle of with an initial velocity . The center of gravity of the arrow moves along the parabola. What is the maximum height h reached by the arrow, and what is its range d? Use the formula:

25.

Simplify and calculate the value of the expression when :

26.

Simplify the expression below hence find its value when .

27.

A number x increases by twelve times the number y. Write down the given expression hence calculate the value of the expression for and .

28.

The product of the numbers and is increased by the sum of the numbers and . Write down the given expression and calculate the value of the expression for and .

29.

The product of and the difference of and is increase by the product of 4 and the sum of and . Write down the given expression. Calculate the value of the expression for and .