Full Grade 6 test

To subtract fractions with different denominators, first find a common denominator.

1. Find the least common multiple of the two denominators.
2. Rewrite both fractions with that common denominator.
3. Subtract the numerators and keep the common denominator.
4. Simplify the resulting fraction. If it is improper, it may be written as a mixed number.

Formula: ^a⁄_b − ^c⁄_d = ^{ad − bc}⁄_bd.

The base angles of an isosceles triangle are equal, and all angles of a triangle add up to 180°.

1. Subtract the given vertex angle from 180°.
2. Divide the remaining angle sum by 2 because the two base angles are equal.

Formula: required angle = (180° − vertex angle) ÷ 2.

Align decimal points and add trailing zeros if needed. Subtract from the right, regrouping one unit of a higher place into ten units of the next place when necessary. Keep the decimal point aligned. For example, 4.2 − 1.35 = 2.85.

Find the place to which you must round and look at the next digit to its right. If it is 0–4, keep the rounding digit; if it is 5–9, increase it by 1. Replace lower whole-number places with zeros.

1 tonne = 1000 kg and 1 kg = 1000 g, so 1 tonne = 1 000 000 g. Multiply tonnes by 1000 for kilograms or by 1 000 000 for grams.

Compare corresponding side ratios. The enlargement factor must be the same for both sides.

Count the terminal branches in each group and add them. Groups may have different numbers of branches.

To find a/b of a number, divide the number by denominator b and multiply by numerator a. For example, 3/4 of 20 is 20 ÷ 4 × 3 = 15.

A full turn around a point is 360°. Add the three known angles and subtract their sum from 360°.

1 litre = 1000 millilitres. Multiply the litres by 1000. For example, 2.5 l = 2500 ml.

The mode is the most frequent data value. Find the greatest frequency in the frequency row, then select the corresponding data value. The answer is the value rather than its frequency.

When one coordinate is shared, the length is the absolute difference of the other coordinates. A distance cannot be negative.

Multiply the aquarium length, width and height to find its volume. Dimensions in decimetres give dm³. Since 1 dm³ = 1 l, the same number gives the water capacity in litres.

Multiply each data value by its frequency and add the products. Divide this sum by the sum of all frequencies. This includes every repeated value.

Find a common denominator, preferably the least common multiple of the denominators. Multiply each numerator and denominator by the same suitable factor. Add the new numerators, keep the common denominator and simplify. For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

One hour contains 60 minutes. Divide the minutes by 60 to obtain hours. For example, 180 min = 180 ÷ 60 = 3 h.

For numbers with the same sign, add their absolute values and keep the sign. For different signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value. For example, −8 + 3 = −5.

Find the second-year harvest by multiplying the first-year amount by (1 + b/100). The third-year increase is based on the second-year harvest, so multiply that amount by (1 + c/100). Finally add the harvests for all three years.

Multiplication by 10 moves the decimal point one place right; multiplication by 100 moves it two places. Add zeros if needed. For example, 2.35 × 10 = 23.5 and 2.35 × 100 = 235.

For store A, add delivery to the price after the percentage discount. For B, subtract the stated euro discount. Compare the total prices.

The quadrant of a point (x, y) depends on the signs of its two coordinates. You do not need to calculate its exact distance from the axes.

1. Look at the first coordinate, x: a positive value means right of the y-axis; a negative value means left.
2. Look at the second coordinate, y: a positive value means above the x-axis; a negative value means below.
3. Combine the two directions and find the corresponding quadrant in the diagram.

Remember: I: (+, +), II: (−, +), III: (−, −), IV: (+, −). In each sign pair, the first sign belongs to x and the second to y.

Quadrants are numbered counterclockwise, starting at the upper right. If either coordinate is 0, the point lies on an axis or at the origin and does not belong to any quadrant.

Subtracting a number means adding its opposite: a − b = a + (−b). Subtracting a negative therefore becomes addition. For example, −4 − (−7) = −4 + 7 = 3.

First calculate the sum in parentheses. Subtracting a negative number means adding its positive opposite.

Rectangular prism volume is V = length × width × height. Express all three lengths in the same units. Lengths in centimetres give volume in cm³.

The equally marked sides AB and AC are equal, so the base angles at B and C are equal. Triangle angles total 180°. The vertex angle CAB = 180° − 2 × angle ABC.

The remainder represents 100% minus the spent percentage. Use that remaining percentage to find the original amount.

Replace division by a fraction with multiplication by its reciprocal: invert the second fraction. Then multiply numerators and denominators and simplify. For example, 2/3 ÷ 4/5 = 2/3 × 5/4 = 5/6.

Solve a × x + b = c using inverse operations: subtract b from c, then divide the difference by a. Thus x = (c − b) ÷ a. Substitute the result into the original equation to check.

Moving right increases the first coordinate; moving down decreases the second. Apply the two changes separately.

First find the larger triangle perimeter. The perimeter ratio of similar triangles equals the ratio of their corresponding sides. Calculate this ratio and multiply the known smaller side length by it.

Rectangle area = length × width. Multiply the two side lengths. If they are in centimetres, express the result in square centimetres, cm².

Add the ratio terms. Divide the triangle total of 180° by that sum, then multiply by the term corresponding to the requested angle.

The range equals the greatest value minus the smallest value. Find these two extremes and subtract; there is no need to add intermediate values.

The mode is the value that occurs most often in a data set.

1. Count how many times each number appears.
2. Find the number with the greatest frequency.

The data does not have to be sorted. The mode is not necessarily the largest number or the mean.

Solve a × x + b = c using inverse operations: subtract b from c, then divide the difference by a. Thus x = (c − b) ÷ a. Substitute the result into the original equation to check.