More Math Lessons

Question 1

Ratios & proportions: Combining Ratios with Different Multipliers - Equate the Common Member

Answer 1

To combine different ratios you must equate the number representing the member common to both ratios.
Write the ratios one on top of the other and expand/reduce the ratios to equate the common member.
Example: Turn this ratio pair

Camels : Goats : Sheep

1st ratio ( 5 : 2 )

2nd ratio ( 1 : 3 )

Into this comparable ratio pair:

Camels : Goats : Sheep

1st ratio ( 5 : 2 )

2nd ratio ( 2 : 6 )

So that the Goats have the same ratio units in both ratios.

Question 2

Ratios & Proportions: Ratio Changes by Addition/Subtraction

Answer 2

A candy box contains only marshmallows and pralines at a ratio of 2:3. A hungry hippo sneaks in and steals 2 marshmallows, leaving the marshmallows and pralines at a new ratio of 4:9. How many pralines are in the box?

When the problem deals with a change in ratio through addition/subtraction i.e., start point ratio, end point ratio, and a real quantity in between:

1) Compare the two ratios - expand / reduce so that the unchanged quantity is represented by the same number in ratio. Example: the original ratio of 2:3 is expanded times 3 to a ratio of 6:9. Note the change in ratio units.

M P

Original ratio 2×3=6 : 3×3=9

change -2 –> -2 marshmallow

New ratio 4 : 9

2) Use the difference in Ratio units and the corresponding change in Real to find the multiplier.

Since a drop of 2 in the marshmallow ratio (from 6 to 4) corresponds to a drop of 2 in real (2 marshmallows stolen by the hungry Hippo), the multiplier is simply ×1.

3) Use the multiplier to find the required quantity. Remember to use the ratios in their expanded/reduced form, rather than the original form.

The question asks for the number of Pralines, which is simply 9×1 = 9.

In Data Sufficiency questions involving ratio changes, a start point ratio, end point ratio, and a real quantity in between are sufficient to answer the question with no calculations needed. In Problem Solving you have to go through all the work.

Question 3

Interest: Simple Interest

Answer 3

For interest questions, Notice the following three pieces of the puzzle:

1) Interest rate - usually given as a %
2) Time - How long the interest is calculated for.
3) Compounded - Annually, semi-annually (every six months), quarterly, monthly.

Simple Interest formula = Principal × Rate × Time

Use simple interest in the following cases:

1) The question explicitly use the phrase “simple interest”
2) The calculation time for the interest is shorter than the stated interest period. e.g. 12% annual interest calculated after a 6-month time period.

Question 4

Interest: Compound Interest

Answer 4

Compound interest is calculated on the principal, as well as on any interest already earned. Use compound interest when the question explicitly uses the term “Compound”.

The compound interest formula is :

Balance = (P+1(r/n))^nt

, where

P = Principal, r = Interest rate (in %), n = number of times per year, t = number of years.

Modeling the problem over a timeline can help. The following timeline is an example of a $10,000 principal which earns 12% annual interest, compounded semiannually:

Question 5

Lines & Angles: Congruent

Answer 5

Congruent means exactly equal in size and shape.

Congruent geometric figures are identical.

Question 6

Triangles: Recycled Right Triangles - The 30°:60°:90° Triangle Practical Use

Answer 6

Triangles: Recycled Right Triangles - The 30°:60°:90° Triangle Practical Use

If a is the smallest leg of a 30°:60°:90° triangle use this ratio box to get you going:

a a/3 2a

30° 60° 90°

Each term in the box is the measure of the side opposite the respective angle.

Question 7

Circles: Tangent Line and Theorem

Answer 7

Circles: Tangent Line and Theorem

A tangent is a line outside the circle, that touches the outer boundary at a single point.
A radius drawn to the point of tangency is always perpendicular to the tangent.

Question 8

Circles: Measuring Arcs

Answer 8

“major arc AB” is the greater of the two parts of the circle that are defined by A and B
“minor arc AB” is the lesser of the two parts of the circle that are defined by A and B
“arc ABC” is the part of the circle on which points A,B,C lie in this order
The measure of the whole circumference is 360°
An arc may be measured in angles, as a fraction of a circle.