How to work out the months for ‘iddah following death or divorce, or fasting for expiation
Question

                My mother is in ‘iddah and it so happens that the three months following my father's death contained only 29 days and the fourth month was complete, i.e., thirty days. Should she add 10 days only, so that it will be four months and 10 days, or should she add 13 days, including three days to make up the shorter months?.
Praise be to Allah.The ‘iddah for a woman whose husband has died is four months and 10 days, because Allaah says (interpretation of the meaning):
“And those of you who die and leave wives behind them, they (the wives) shall wait (as regards their marriage) for four months and ten days, then when they have fulfilled their term”
[al-Baqarah 2:234] 
This period begins when the husband dies and ends when that period ends. 
Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (8/93):
The scholars are unanimously agreed that the ‘iddah of a free Muslim women who is not pregnant lasts for four months and 10 days from the death of her husband, regardless of whether the marriage was consummated with her or not, whether she was an adult or a minor, because Allaah says (interpretation of the meaning):
“And those of you who die and leave wives behind them, they (the wives) shall wait (as regards their marriage) for four months and ten days, then when they have fulfilled their term”
[al-Baqarah 2:234].
And the Prophet (blessings and peace of Allaah be upon him) said: “It is not permissible for a woman who believes in Allaah and the Last Day to mourn for anyone who dies for more than three days, except for a husband, four months and 10 days.” Agreed upon. 
She should count the Arabic, lunar months and not the days, according to the opinion of the majority of fuqaha’, whether the month is complete or not. When she has completed four months, she should add 10 days of the fifth month, and thus she will have completed her ‘iddah. 
This applies if the death occurs on the first of the month. If he dies during the month, then she should count the rest of the first month and three months with their new moons -- whether the months are complete or not -- and 10 days. With regard to what she missed of the first month, she may count it in two ways according to the scholars:
1.     She may regard the month as 30 days, whether it turned out to be complete (30 days) or incomplete (29 days);
2.     She may count as ‘iddah the same amount of time in the fifth month as she missed in the first month. If the first month turned out to be complete, then she should count the number of days to complete 30 days, and if it turns out to be incomplete, then she should count the number of days to complete 29 days. 
See: al-Mughni, 8/85; Kashshaaf al-Qinaa’, 5/418; al-Mawsoo’ah al-Fiqhiyyah, 29/315 
The second view was favoured by Shaykh al-Islam Ibn Taymiyah, and was regarded as more correct by our contemporary scholar Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him). 
Shaykh Ibn ‘Uthaymeen said with regard to one who fasts for two consecutive months starting from the 15th of Jumaada al-Oola, and both Jumaada al-Oola and Jumaada al-Awwal have 29 days, that he should end his fast by fasting on the 15th of Rajab, according to the view that he should complete the first month as 30 days. 
But according to the more correct opinion, the two months are reckoned according to the new moons, so he should end his fast by fasting the 14th day of Rajab. 
Al-Sharh al-Mumti’, 6/413, 414 
Similarly, Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said that there is no need for us to count the number of days, rather we should look at the day when he began in the first month, and the end will be the same day in the other month. End quote from Majmoo’ al-Fataawa, 25/144. 
An example of that in the case of death: if a man dies on the 12th of Muharram, his wife should observe ‘iddah until the 12th of Jumaada al-Awwal. This is four months, regardless of whether they are complete or not. Then she should add 10 days, so her ‘iddah ends on the 22nd of Jumaada al-ِِAwwal at the time when her husband died. 
Based on that, all your mother has to do is add 10 days only, and she does not have to complete the incomplete months to make them 30 days. 
What we have said here about the ‘iddah following the husband's death also applies to the one who fasts two consecutive months, and it also applies to the ‘iddah of divorce if it is counted in months, which is the case with regard to a divorced woman who is a minor or a woman past the age of bearing children who does not menstruate. 
And Allaah knows best.