This page focuses on solving changing ratio problems with algebra, providing a comprehensive example and step-by-step solution process.

The example problem involves Jan and Kim's marbles, initially in a ratio of 5:6. After Jan gains 2 marbles, the new ratio becomes 7:8. The goal is to determine how many marbles each owned initially.

Example: Jan and Kim own numbers of marbles that are in the ratio 5:6. Jan gains 2 more marbles and the ratio is now 7:8. How many marbles do each own initially?

The solution process involves:

1. Writing the initial ratio in algebraic form (5x : 6x)
2. Expressing the new ratio after the change $5x+2 : 6x$
3. Setting up an equation based on the new ratio $5x+2 : 6x = 7 : 8$
4. Using cross multiplication to solve for x
5. Substituting the value of x to find the original numbers of marbles

Highlight: This problem demonstrates how to apply algebraic techniques to solve complex ratio problems involving changes in quantities.

The solution reveals that initially, Jan owned 40 marbles and Kim owned 48 marbles.

Vocabulary: Changing ratios refer to situations where the ratio between quantities changes due to an increase or decrease in one or more of the quantities involved.

This example illustrates the power of algebra in solving ratio problems, especially when dealing with changing ratios. It combines the concepts of cross multiplication, algebraic manipulation, and ratio analysis to arrive at a precise solution.

Quote: "Initially Jan owned 40, Kim owned 48"

This page provides valuable practice for students preparing for exams like GCSE, where changing ratio questions and algebraic ratio GCSE questions are common.

This page introduces the concept of using cross multiplication to solve ratio problems with variables. It provides a step-by-step example of how to apply this method effectively.

The example problem involves two brothers, Rich and Andrew, sharing money in a ratio of 2:7, with Andrew receiving £30 more than Rich. The solution demonstrates how to use blocks to represent the ratio and solve for the total amount shared.

Example: Two brothers, Rich and Andrew, share a sum of money in the ratio 2:7. Andrew gets £30 more than Rich. Calculate how much the brothers share.

The solution process involves:

1. Representing the difference between the ratio parts $7 - 2 = 5$
2. Assigning a value to the difference (5 blocks = £30)
3. Calculating the value of one block (£6)
4. Determining the total amount shared $6 × (7 + 2) = £54$

Highlight: This method of using blocks to represent ratio parts is particularly useful for visualizing and solving ratio problems.

The page also covers finding the value of n in a ratio, using the example $n+3$ : 2n = 3 : 5. This introduces the concept of cross multiplication for solving proportions with variables.

Vocabulary: Cross multiplication is a technique used to solve proportions by multiplying the numerator of each fraction by the denominator of the other fraction.

The solution process for finding n involves:

1. Cross multiplying the ratios
2. Simplifying the resulting equation
3. Solving for n

Definition: The cross multiplication property of proportion states that if a:b = c:d, then ad = bc.