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You have a total of 21 coins, all nickels and dimes. The total value is $1.70. How many nickels and how many dimes do you have?

Last updated date: 16th Jul 2024
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Hint:To solve this question, you need to know about the relation between nickel, dimes, cents and dollars. Here is the relation between them as follows
Five cents equal one nickel and one dime equals ten cents, whereas a hundred cents equals a dollar.
First of all take two variables for nickel and dimes and also convert the given value into cents, then with the help of the given information and relation write two equations and solve them to get the required solution.

Complete step by step solution:
Let us represent the number of nickels and dimes as n and d respectively.
$5{\text{cents}} = 1{\text{nickel}},\;10{\text{cents}} = 1{\text{dime}}\;{\text{and}}\;100{\text{cents}}
= 1\$ $
We have n number of nickels and d number of dimes, and a total of 21 coins, we can write this as
n + d = 21 \\
n = 21 - d - - - - - - (i) \\
Also we can write $\$ 1.70 = 100 \times 1.70{\text{cents}} = 170{\text{cents}}$
Another equation from given information can be written as
$5n + 10d = 170$
We have converted nickels and dimes to cents,
Substituting value of n from equation (i), we will get
\Rightarrow 5(21 - d) + 10d = 170 \\
\Rightarrow 5 \times 21 - 5d + 10d = 170 \\
\Rightarrow 105 + 5d = 170 \\
\Rightarrow 5d = 170 - 105 \\

\Rightarrow 5d = 65 \\
\Rightarrow d = \dfrac{{65}}{5} \\
\Rightarrow d = 13 \\
Now putting value of d in equation (i), to get the value of n,
\Rightarrow n = 21 - 13 \\
\Rightarrow n = 8 \\
Therefore we have 8 nickels and 13 dimes.

Note: When writing the equation, make sure you have converted all the terms into similar units because all types of mathematical operations only occur between similar terms or units. Also remembering some basic conversions between units is important as well as helpful in solving questions which have quantities with different-different units.