Algebra

Linear Equation

Medium

A basket of $2$ apples, $4$ oranges and $6$ mangoes costs the same as a basket of $1$ apple, $4$ oranges and $8$ mangoes, or a basket of $8$ oranges and $7$ mangoes. Then the number of mangoes in a basket of mangoes that has the same cost as the other baskets is

$12$

$13$

$10$

$11$

✅ Correct Option: 2

We have three different fruit baskets that all cost the same amount. We need to find how many mangoes alone would equal this same cost.

Let's use variables to represent the cost of each fruit:

A = cost of 1 apple

O = cost of 1 orange

M = cost of 1 mango

Since all three baskets cost the same, we can write:

Basket 1 = Basket 2:

$2A + 4O + 6M = 1A + 4O + 8M$

Basket 2 = Basket 3:

$1A + 4O + 8M = 8O + 7M$

From the first equation:

$2A + 4O + 6M = 1A + 4O + 8M$

$2A + 6M = 1A + 8M$

$1A + 6M = 8M$

$1A = 2M$

One apple costs the same as 2 mangoes.

From the second equation:

$1A + 4O + 8M = 8O + 7M$

Substituting $A = 2M$ :

$2M + 4O + 8M = 8O + 7M$

$10M + 4O = 8O + 7M$

$10M = 4O + 7M$

$3M = 4O$

3 mangoes cost the same as 4 oranges.

Now we can express the cost of any basket in terms of mangoes only.

Taking the first basket: $2A + 4O + 6M$

Replace apples with mangoes using $A = 2M$ :

$2A = 2 \times 2M = 4M$

Replace oranges with mangoes using $3M = 4O$ :

$4O = 3M$

$2A + 4O + 6M = 4M + 3M + 6M = 13M$

A basket containing only mangoes that costs the same as the other baskets would contain 13 mangoes.

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